7483:
6982:
7478:{\displaystyle {\begin{matrix}{\begin{array}{|c||c|c|c|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{2}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {2}}\\&&2&\\3&4&&\\\hline \end{array}}&\quad {\begin{aligned}f(x)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac {(x-1)(x-2)}{2}}\\\\&=2+(x-1)(x-2)\\\end{aligned}}\end{matrix}}}
11910:
329:
11244:
1625:
11905:{\displaystyle {\begin{aligned}f_{x}(x,y)&\approx {\frac {f(x+h,y)-f(x-h,y)}{2h}}\\f_{y}(x,y)&\approx {\frac {f(x,y+k)-f(x,y-k)}{2k}}\\f_{xx}(x,y)&\approx {\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\\f_{yy}(x,y)&\approx {\frac {f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}}\\f_{xy}(x,y)&\approx {\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}.\end{aligned}}}
10666:
7742:
8628:
8793:
6063:
3623:
2627:
2347:
2062:
9988:
10495:
9141:
10507:
9495:
7498:
5799:
of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid. This is useful for differentiating a function on a grid, where, as one approaches the edge of
8647:
5794:
Using linear algebra one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the
9773:
8993:
8397:
10959:
6534:
5819:
3355:
6242:
3109:
12179:
10343:
3812:
9025:
9316:
8125:
10765:
968:
9812:
2796:
2356:
2076:
1803:
1443:
1331:
1610:
Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).
9604:
8857:
10777:
7843:
2939:
10661:{\displaystyle \ \nabla \left({\frac {f}{g}}\right)=\left.\left(\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\right)\right/\left(g\cdot \det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}\right)}
5800:
the grid, one must sample fewer and fewer points on one side. Finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order may be constructed.
271:, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives". Finite difference approximations are finite difference quotients in the terminology employed above.
6932:
1550:
7737:{\displaystyle \Delta _{j,0}=y_{j},\qquad \Delta _{j,k}={\frac {\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_{j}}}\quad \ni \quad \left\{k>0,\;j\leq \max \left(j\right)-k\right\},\qquad \Delta 0_{k}=\Delta _{0,k}}
8208:
10132:
11083:
8623:{\displaystyle \ \operatorname {\Delta } _{h}{\bigl (}\ f(x)\ g(x)\ {\bigr )}\ =\ {\bigl (}\ \operatorname {\Delta } _{h}f(x)\ {\bigr )}\ g(x+h)\ +\ f(x)\ {\bigl (}\ \operatorname {\Delta } _{h}g(x)\ {\bigr )}~.}
1203:
5251:
4842:
1095:
9585:
6073:
6335:
4658:
10311:
8788:{\displaystyle \operatorname {\Delta } _{h}=h\operatorname {D} +{\frac {1}{2!}}h^{2}\operatorname {D} ^{2}+{\frac {1}{3!}}h^{3}\operatorname {D} ^{3}+\cdots =e^{h\operatorname {D} }-\operatorname {I} \ ,}
772:
11921:
10348:
3321:
3238:
7905:
4270:
3634:
613:
6668:
11249:
10782:
7955:
7134:
5442:
4415:
10671:
6058:{\displaystyle \Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-1}\cdots \sum \limits _{i_{n}=0}^{k-1}\Delta _{h}^{n}\left(f,x+i_{1}h+i_{2}h+\cdots +i_{n}h\right).}
787:
6765:
5617:
2948:
1721:
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for
3627:
Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order
4064:
3977:
495:
9002:. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to
5042:
3618:{\displaystyle {\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}(x)}{h^{n}}}+o(h)={\frac {\nabla _{h}^{n}(x)}{h^{n}}}+o(h)={\frac {\delta _{h}^{n}(x)}{h^{n}}}+o\left(h^{2}\right).}
238:
5773:
6991:
9983:{\displaystyle \ {\frac {\Delta _{h}}{h}}(1+\lambda h)^{\frac {x}{h}}={\frac {\Delta _{h}}{h}}e^{\ln(1+\lambda h){\frac {x}{h}}}=\lambda e^{\ln(1+\lambda h){\frac {x}{h}}}\ ,}
10209:
3160:
multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied substituting the average of
404:
10490:{\displaystyle {\begin{aligned}\ \Delta (fg)&=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g\\\nabla (fg)&=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g\ \end{aligned}}}
6598:; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Yet clearly, the sine function is not zero.) Here, the expression
2622:{\displaystyle f''(x)\approx {\frac {\nabla _{h}^{2}(x)}{h^{2}}}={\frac {{\frac {f(x)-f(x-h)}{h}}-{\frac {f(x-h)-f(x-2h)}{h}}}{h}}={\frac {f(x)-2f(x-h)+f(x-2h)}{h^{2}}}.}
2342:{\displaystyle f''(x)\approx {\frac {\Delta _{h}^{2}(x)}{h^{2}}}={\frac {{\frac {f(x+2h)-f(x+h)}{h}}-{\frac {f(x+h)-f(x)}{h}}}{h}}={\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}.}
1338:
1226:
2648:
9257:
9136:{\displaystyle h\operatorname {D} =-\ln(1-\nabla _{h})\quad {\text{ and }}\quad h\operatorname {D} =2\operatorname {arsinh} \left({\tfrac {1}{2}}\,\delta _{h}\right)~.}
7747:
2057:{\displaystyle f''(x)\approx {\frac {\delta _{h}^{2}(x)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.}
161:
124:
9490:{\displaystyle \ (x)_{n}\equiv \left(\ x\ \operatorname {T} _{h}^{-1}\right)^{n}=x\left(x-h\right)\left(x-2h\right)\cdots {\bigl (}x-\left(n-1\right)\ h{\bigr )}\ ,}
6596:
6576:
6810:
3345:
techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large
8158:
6274:. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called
10018:
1454:
13918:
10980:
1106:
5178:
4720:
9500:
2805:
4439:
287:
13906:
9997:, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials. This umbral exponential thus amounts to the exponential
10233:
7934:
provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist.
1002:
656:
5171:, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree
9768:{\displaystyle \ \sin \left(x\ \operatorname {T} _{h}^{-1}\right)=x-{\frac {(x)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(x)_{7}}{7!}}+\cdots \ }
12966:
7852:
257:
8988:{\displaystyle h\operatorname {D} =\ln(1+\Delta _{h})=\Delta _{h}-{\tfrac {1}{2}}\,\Delta _{h}^{2}+{\tfrac {1}{3}}\,\Delta _{h}^{3}-\cdots ~.}
6325:
4109:
510:
14028:
13913:
6601:
10954:{\displaystyle {\begin{aligned}\ \sum _{n=a}^{b}\Delta f(n)&=f(b+1)-f(a)\\\sum _{n=a}^{b}\nabla f(n)&=f(b)-f(a-1)\ \end{aligned}}}
6529:{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}(a)}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}(a),}
5384:
4324:
13896:
13891:
6677:
5562:
12973:
14059:
13901:
13886:
13000:
12283:
13188:
6263:
13881:
8388:
6945:
Newton series expansions can be superior to Taylor series expansions when applied to discrete quantities like quantum spins (see
6067:
3997:
3894:
416:
3243:
3163:
1689:
14069:
11918:
is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is
6970:
that reproduces these values, by first computing a difference table, and then substituting the differences that correspond to
1661:
14074:
13498:
13252:
12875:
12675:
12388:
12358:
12325:
6946:
4961:
166:
12247:
5717:
3333:
of the sequence, and have a number of interesting combinatorial properties. Forward differences may be evaluated using the
1554:
The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If
6237:{\displaystyle \Delta _{h}^{n}(fg,x)=\sum \limits _{k=0}^{n}{\binom {n}{k}}\Delta _{h}^{k}(f,x)\Delta _{h}^{n-k}(g,x+kh).}
3838:
If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.
3337:. The integral representation for these types of series is interesting, because the integral can often be evaluated using
1103:
has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written
6785:
is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of
1668:
256:. There are many similarities between difference equations and differential equations, specially in the solving methods.
9587:
hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function
3104:{\displaystyle \delta _{h}^{n}(x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right).}
13050:
12929:
12412:
1642:
17:
6281:
Common applications of the finite difference method are in computational science and engineering disciplines, such as
13996:
13855:
12840:
12815:
5275:
Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again:
1708:
12174:{\displaystyle f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)+f(x-h,y-k)}{2hk}},}
10179:
13410:
13326:
1675:
14084:
13991:
13923:
13548:
13403:
13371:
13130:
10770:
3807:{\displaystyle {\frac {\Delta _{h}(x)-{\frac {1}{2}}\Delta _{h}^{2}(x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}}
12943:
6956:
To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the
14064:
13624:
13601:
13316:
6267:
1646:
1657:
13714:
13652:
13447:
13321:
12993:
12953:
6320:
6271:
5783:
8120:{\displaystyle f(x)=\sum _{k=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}f(a+jh).}
5710:
Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant
13200:
13178:
10760:{\displaystyle \nabla \left({\frac {f}{g}}\right)={\frac {g\,\nabla f-f\,\nabla g}{g\cdot (g-\nabla g)}}\ }
10214:
8822:; in the special case that the series of derivatives terminates (when the function operated on is a finite
6804:
963:{\displaystyle \delta _{h}(x)=f(x+{\tfrac {h}{2}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}(x)+\nabla _{h/2}(x).}
14023:
12915:
3334:
14079:
14054:
14008:
13774:
13388:
13210:
12948:
12960:
12424:
3352:
The relationship of these higher-order differences with the respective derivatives is straightforward,
13393:
13163:
8995:
This formula holds in the sense that both operators give the same result when applied to a polynomial.
6950:
13812:
13759:
12263:
1596:
978:
13220:
13928:
13699:
13247:
12986:
6275:
6253:
4068:
Only the coefficient of the highest-order term remains. As this result is constant with respect to
2791:{\displaystyle \Delta _{h}^{n}(x)=\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f{\bigl (}x+ih{\bigr )},}
84:
4854:
This identity can be used to find the lowest-degree polynomial that intercepts a number of points
370:
13694:
13366:
12439:
12181:
since the only values to compute that are not already needed for the previous four equations are
9262:
A large number of formal differential relations of standard calculus involving functions
8998:
Even for analytic functions, the series on the right is not guaranteed to converge; it may be an
1635:
1447:
However, the central (also called centered) difference yields a more accurate approximation. If
260:
can be written as difference equations by replacing iteration notation with finite differences.
13822:
13704:
13525:
13473:
13279:
13257:
13125:
10227:
10173:
9166:
8135:
5381:
Here, the constant is achieved after only two pairwise differences, thus the following result:
1682:
407:
127:
96:
12317:
12310:
5714:. Thus, the lowest-degree polynomial intercepting all the points in the first table is found:
4942:, the following relation to the cells in the column immediately to the left exists for a cell
278:
in 1715 and have also been studied as abstract self-standing mathematical objects in works by
13948:
13807:
13719:
13376:
13311:
13284:
13274:
13195:
13183:
13168:
13140:
12856:
11196:
11176:
10158:
10145:
The inverse operator of the forward difference operator, so then the umbral integral, is the
10139:
249:
137:
109:
92:
12665:
12513:
13764:
13383:
13230:
12770:
12713:
12587:
12253:
11111:
10009:
7931:
6671:
3338:
3142:
3115:
1604:
12403:
Jordán, op. cit., p. 1 and Milne-Thomson, p. xxi. Milne-Thomson, Louis
Melville (2000):
9149:
of combinatorics. This remarkably systematic correspondence is due to the identity of the
6581:
6561:
8:
13784:
13709:
13596:
13553:
13304:
13289:
13120:
13108:
13095:
13055:
13035:
12622:
12501:
12278:
12242:
10135:
9998:
9995:
Fourier sums of continuum functions are readily, faithfully mapped to umbral
Fourier sums
9310:
8139:
7917:
7492:
6800:
6282:
1216:
997:
283:
245:
241:
103:
75:
12879:
12774:
12717:
12591:
11192:
292:
13873:
13848:
13679:
13632:
13573:
13538:
13533:
13513:
13508:
13503:
13468:
13415:
13398:
13299:
13173:
13158:
13103:
13070:
12788:
12760:
12729:
12703:
12605:
12577:
12544:
12273:
11235:
7952:
In a compressed and slightly more general form and equidistant nodes the formula reads
6957:
6259:
5789:
3330:
264:
88:
11234:
Finite differences can be considered in more than one variable. They are analogous to
11211:
14013:
13837:
13769:
13591:
13568:
13442:
13435:
13338:
13153:
12925:
12898:
12836:
12811:
12671:
12609:
12486:
12408:
12384:
12354:
12321:
12237:
11203:
10002:
8999:
8815:
8263:
7946:
6768:
6548:
6299:
12792:
12733:
1438:{\displaystyle {\frac {\nabla _{h}(x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}
1326:{\displaystyle {\frac {\Delta _{h}(x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.}
977:
Finite difference is often used as an approximation of the derivative, typically in
13971:
13754:
13667:
13647:
13578:
13488:
13430:
13422:
13356:
13269:
13030:
13025:
12894:
12778:
12721:
12649:
12595:
12451:
12268:
6935:
6556:
5796:
312:. The formal calculus of finite differences can be viewed as an alternative to the
12600:
12565:
12529:
14033:
14018:
13802:
13657:
13637:
13606:
13583:
13563:
13457:
13113:
13060:
12497:
9778:
9146:
8336:
7938:
7838:{\displaystyle {P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\xi -x_{k}\right),}
6939:
6286:
2934:{\displaystyle \nabla _{h}^{n}(x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),}
13943:
13842:
13689:
13642:
13543:
13346:
12643:
12346:
10146:
8221:
7846:
13361:
12725:
12455:
4936:
We can use a differences table, where for all cells to the right of the first
14048:
13817:
13672:
13558:
13262:
13237:
12691:
10500:
10013:
9022:
The analogous formulas for the backward and central difference operators are
8637:
8269:
The finite difference of higher orders can be defined in recursive manner as
7942:
7910:
6772:
6578:. This is easily seen, as the sine function vanishes at integer multiples of
4693:
as the coefficient of the highest-order term. Given the assumption above and
3832:
1790:, we obtain the central difference approximation of the second derivative of
317:
12880:"Mellin transforms and asymptotics: Finite differences and Rice's integrals"
12783:
12748:
7949:, all of which are defined in terms of suitably scaled forward differences.
7924:
is a polynomial function can be weakened all the way to the assumption that
298:
13827:
13797:
13662:
13225:
12639:
12502:" Why don't they teach Newton's calculus of 'What comes next?' " on YouTube
10336:
10316:
All of the above rules apply equally well to any difference operator as to
6927:{\displaystyle (x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},}
6316:
6308:
3342:
328:
309:
279:
275:
11110:
is a further generalization, where the finite sum above is replaced by an
2067:
Similarly we can apply other differencing formulas in a recursive manner.
13075:
13017:
12530:"Three notes on Ser's and Hasse's representations for the zeta-functions"
12258:
11207:
9150:
11206:, difference operators and other Möbius inversion can be represented by
8203:{\displaystyle \Delta _{h}=\operatorname {T} _{h}-\operatorname {I} \ ,}
6329:
in 1687, namely the discrete analog of the continuous Taylor expansion,
332:
The three types of the finite differences. The central difference about
13792:
13724:
13478:
13351:
13215:
13205:
13148:
8823:
6967:
6540:
3847:
1649: in this section. Unsourced material may be challenged and removed.
985:
253:
80:
10138:
can often be solved with techniques very similar to those for solving
10127:{\displaystyle \ \delta (x)\mapsto {\frac {\sin \left}{\pi (x+h)}}\ ,}
1545:{\displaystyle {\frac {\delta _{h}(x)}{h}}-f'(x)=o\left(h^{2}\right).}
13986:
13734:
13729:
13040:
12961:
Table of useful finite difference formula generated using
Mathematica
11078:{\displaystyle \Delta _{h}^{\mu }(x)=\sum _{k=0}^{N}\mu _{k}f(x+kh),}
8633:
Similar
Leibniz rules hold for the backward and central differences.
1198:{\displaystyle {\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}(x)}{h}}.}
10152:
5246:{\displaystyle 648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162}
4837:{\displaystyle \Delta _{h}^{m-1}(x)=ahm\cdot h^{m-1}(m-1)!=ah^{m}m!}
1624:
13981:
13483:
13009:
12582:
12549:
12475:
12307:
6938:), are included in the observations that matured to the system of
3835:, or by using the calculus of finite differences, explained below.
3326:
313:
12765:
12708:
8799:
denotes the conventional, continuous derivative operator, mapping
5559:
Thus the constant is achieved after only one pairwise difference:
4314:. Assuming the following holds true for all polynomials of degree
3987:
pairwise differences, the following result can be achieved, where
336:
gives the best approximation of the derivative of the function at
248:
that involves the finite difference operator in the same way as a
13832:
13085:
9580:{\displaystyle \ {\frac {\Delta _{h}}{h}}(x)_{n}=n\ (x)_{n-1}\ ,}
12312:
The
Mathematics of Financial Derivatives: A Student Introduction
5466:
Moving on to the next term, by subtracting out the second term:
4653:{\displaystyle \Delta _{h}(x)=-=ahmx^{m-1}+{\text{l.o.t.}}=T(x)}
1779:
and applying a central difference formula for the derivative of
14001:
13065:
11182:
The generalized difference can be seen as the polynomial rings
10306:{\displaystyle \ \Delta (a\ f+b\ g)=a\ \Delta f+b\ \Delta g\ }
1215:
is small. The error in this approximation can be derived from
297:(1939). Finite differences trace their origins back to one of
13080:
11175:. Such generalizations are useful for constructing different
1090:{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}
12537:
Integers (Electronic
Journal of Combinatorial Number Theory)
11914:
Alternatively, for applications in which the computation of
767:{\displaystyle \nabla _{h}(x)=f(x)-f(x-h)=\Delta _{h}(x-h).}
12978:
10596:
10539:
3831:. This can be proven by expanding the above expression in
3316:{\displaystyle \ \delta ^{n}(\ x+{\tfrac {\ h\ }{2}}\ )~.}
3233:{\displaystyle \ \delta ^{n}(\ x-{\tfrac {\ h\ }{2}}\ )\ }
7900:{\displaystyle f(\xi )=\Delta 0\cdot P\left(\xi \right).}
5046:
To find the first term, the following table can be used:
12863:. Englewood Cliffs, NJ: Prentice-Hall. Section 2.2.
4265:{\displaystyle \Delta _{h}(x)=Q(x+h)-Q(x)=-=ah=ah^{1}1!}
12566:"Newton series expansion of bosonic operator functions"
12527:
11114:. Another way of generalization is making coefficients
4074:, any further pairwise differences will have the value
12974:
Discrete Second
Derivative from Unevenly Spaced Points
12694:(2008). "Umbral deformations on discrete space-time".
12353:. Springer Science & Business Media. p. 182.
10619:
10554:
9275:
systematically map to umbral finite-difference analogs
9100:
8946:
8915:
6987:
3282:
3202:
865:
835:
608:{\displaystyle \Delta (x)=\Delta _{1}(x)=f(x+1)-f(x).}
12912:
12440:"On the Graphic Delineation of Interpolation Formulæ"
11924:
11247:
10983:
10780:
10674:
10510:
10346:
10236:
10182:
10021:
9815:
9607:
9503:
9319:
9169:
9153:
of the umbral quantities to their continuum analogs (
9145:
The calculus of finite differences is related to the
9028:
8860:
8650:
8400:
8161:
7958:
7855:
7750:
7501:
7378:
7349:
7250:
7189:
6985:
6813:
6680:
6663:{\displaystyle {\binom {x}{k}}={\frac {(x)_{k}}{k!}}}
6604:
6584:
6564:
6338:
6258:
An important application of finite differences is in
6076:
5822:
5720:
5565:
5387:
5181:
4964:
4723:
4442:
4327:
4112:
4000:
3897:
3637:
3358:
3246:
3166:
2951:
2808:
2651:
2359:
2079:
1806:
1457:
1341:
1229:
1109:
1005:
790:
659:
513:
419:
373:
169:
140:
112:
12833:
Numerical
Methods for Partial Differential Equations
12303:
12301:
12299:
9309:
is a generalization of the above falling factorial (
3994:
is a real number marking the arithmetic difference:
1599:. This is particularly troublesome if the domain of
1335:
The same formula holds for the backward difference:
5437:{\displaystyle -306=a\cdot 3^{2}\cdot 2!=a\cdot 18}
4410:{\displaystyle \Delta _{h}^{m-1}(x)=ah^{m-1}(m-1)!}
12968:Finite Calculus: A Tutorial for Solving Nasty Sums
12835:. New York, NY: Academic Press. Section 1.6.
12648:(2nd ed.). Macmillan and Company – via
12309:
12173:
11904:
11229:
11077:
10953:
10759:
10660:
10489:
10305:
10203:
10126:
9982:
9767:
9579:
9489:
9251:
9135:
8987:
8787:
8622:
8202:
8119:
7899:
7837:
7736:
7487:For the case of nonuniform steps in the values of
7477:
6926:
6759:
6662:
6590:
6570:
6528:
6236:
6057:
5767:
5611:
5436:
5245:
5036:
4836:
4652:
4409:
4264:
4058:
3971:
3806:
3617:
3315:
3232:
3103:
2933:
2790:
2621:
2341:
2056:
1544:
1437:
1325:
1197:
1089:
962:
766:
607:
489:
398:
232:
155:
118:
12873:
12296:
10153:Rules for calculus of finite difference operators
8129:
8087:
8074:
8019:
7993:
6876:
6863:
6799:Note the formal correspondence of this result to
6760:{\displaystyle (x)_{k}=x(x-1)(x-2)\cdots (x-k+1)}
6621:
6608:
6488:
6467:
6150:
6137:
5612:{\displaystyle 108=a\cdot 3^{1}\cdot 1!=a\cdot 3}
4954:, with the top-leftmost cell being at coordinate
4307:and the coefficient of the highest-order term be
3044:
3031:
2901:
2888:
2750:
2737:
14046:
12645:A Treatise on the Calculus of Finite Differences
12378:
12308:Paul Wilmott; Sam Howison; Jeff Dewynne (1995).
11202:As a convolution operator: Via the formalism of
10611:
10546:
7673:
1027:
12746:
8854:and formally inverting the exponential yields
8134:The forward difference can be considered as an
4870:-axis from one point to the next is a constant
3152:Note that the central difference will, for odd
12374:
12372:
12370:
12351:Introduction to Partial Differential Equations
9305:For instance, the umbral analog of a monomial
8814:The expansion is valid when both sides act on
4700:pairwise differences (resulting in a total of
268:
83:by finite differences plays a central role in
12994:
12922:Difference Equations: Theory and Applications
12627:Difference Methods for Initial Value Problems
9476:
9438:
9015:
8609:
8569:
8517:
8477:
8461:
8421:
6934:(following from it, and corresponding to the
5628:and thus the third term of the polynomial is
2780:
2761:
11241:Some partial derivative approximations are:
6247:
5777:
5265:. Thus, the first term of the polynomial is
972:
503:may be variable or constant. When omitted,
12861:Finite-Difference Equations and Simulations
12805:
12367:
12345:
12341:
12339:
12337:
9806: also happens to be an exponential,
4877:. For example, given the following points:
4059:{\displaystyle \Delta _{h}^{n}(x)=ah^{n}n!}
3972:{\displaystyle P(x)=ax^{n}+bx^{n-1}+l.o.t.}
3145:provides the coefficient for each value of
1614:
490:{\displaystyle \Delta _{h}(x)=f(x+h)-f(x).}
344:Three basic types are commonly considered:
13001:
12987:
12855:
12563:
7666:
6979:(underlined) into the formula as follows,
499:Depending on the application, the spacing
14029:Regiomontanus' angle maximization problem
12782:
12764:
12707:
12696:International Journal of Modern Physics A
12599:
12581:
12548:
12487:Finite Difference Coefficients Calculator
12284:Upwind differencing scheme for convection
11210:with a function on the poset, called the
10718:
10705:
10473:
10457:
10444:
10408:
10392:
10379:
9198:
9111:
8957:
8926:
6904:
6494:
6417:
6266:, which aim at the numerical solution of
1709:Learn how and when to remove this message
1207:Hence, the forward difference divided by
267:, finite differences are widely used for
36:is a mathematical expression of the form
13872:
12747:Curtright, T. L.; Zachos, C. K. (2013).
12334:
2639:differences are given by, respectively,
2637:-th order forward, backward, and central
327:
13377:Differentiating under the integral sign
5549:−2424 − (−17(13)) = −2424 + 2873 = 449
5536:−1359 − (−17(10)) = −1359 + 1700 = 341
5037:{\displaystyle (a+1,b+1)=(a,b+1)-(a,b)}
233:{\displaystyle \Delta (x)=f(x+1)-f(x).}
63:. If a finite difference is divided by
14:
14047:
12690:
12663:
12437:
12316:. Cambridge University Press. p.
10012:maps to its umbral correspondent, the
6803:. Historically, this, as well as the
274:Finite differences were introduced by
13253:Inverse functions and differentiation
12982:
12638:
12557:
12444:Journal of the Institute of Actuaries
12438:Fraser, Duncan C. (January 1, 1909).
9598: in such symbols), and so on.
7937:The Newton series, together with the
5768:{\displaystyle 4x^{3}-17x^{2}+36x-19}
12830:
12654:Also, a Dover reprint edition, 1960.
12248:Finite-difference time-domain method
9013:mentioned at the end of the section
7845:and the resulting polynomial is the
6313:Gregory–Newton interpolation formula
5261:, it can be found to have the value
1647:adding citations to reliable sources
1618:
12564:König, Jürgen; Hucht, Fred (2021).
12425:"Finite differences of polynomials"
11191:Difference operator generalizes to
7945:, is a special case of the general
6771:" or "lower factorial", while the
6114:
5931:
5894:
5860:
5523:−600 − (−17(7)) = −600 + 833 = 233
24:
13051:Free variables and bound variables
12405:The Calculus of Finite Differences
11188:. It leads to difference algebras.
10985:
10967:
10889:
10809:
10742:
10719:
10706:
10675:
10627:
10565:
10557:
10514:
10474:
10467:
10458:
10445:
10419:
10409:
10402:
10393:
10380:
10354:
10294:
10279:
10240:
10186:
9872:
9822:
9629:
9510:
9358:
9228:
9200:
9178:
9079:
9057:
9032:
8959:
8928:
8902:
8886:
8864:
8773:
8765:
8739:
8701:
8669:
8653:
8579:
8487:
8406:
8188:
8176:
8163:
8078:
7997:
7871:
7719:
7702:
7589:
7558:
7536:
7503:
7237:
7176:
7157:
7030:
7018:
7006:
6867:
6612:
6496:
6471:
6459:
6379:
6370:
6305:Newton forward difference equation
6292:
6187:
6157:
6141:
6078:
5965:
5824:
5637:. Subtracting out the third term:
5510:−147 − (−17(4)) = −147 + 272 = 125
5456:, the polynomial's second term is
5368:6364 − 4(13) = 6364 − 8788 = −2424
5352:2641 − 4(10) = 2641 − 4000 = −1359
5175:. Thus, using the identity above:
4725:
4444:
4329:
4274:This proves it for the base case.
4114:
4081:
4002:
3683:
3642:
3476:
3409:
3118:after the summation sign shown as
3035:
2892:
2810:
2741:
2653:
2384:
2104:
1346:
1234:
1159:
922:
883:
728:
661:
539:
514:
421:
375:
170:
141:
113:
25:
14096:
13856:The Method of Mechanical Theorems
12936:
8305:Another equivalent definition is
6953:or discrete counting statistics.
6947:Holstein–Primakoff transformation
4277:
3325:Forward differences applied to a
1223:is twice differentiable, we have
1211:approximates the derivative when
14060:Numerical differential equations
13411:Partial fractions in integration
13327:Stochastic differential equation
12528:Iaroslav V. Blagouchine (2018).
10159:rules for finding the derivative
9601:For example, the umbral sine is
7920:states that the assumption that
6264:numerical differential equations
4436:. With one pairwise difference:
1623:
13549:Jacobian matrix and determinant
13404:Tangent half-angle substitution
13372:Fundamental theorem of calculus
12867:
12849:
12824:
12799:
12740:
12684:
12657:
12632:
12616:
12521:
12506:
11230:Multivariate finite differences
11217:; for the difference operator,
9075:
9069:
9016:§ Higher-order differences
8826:) the expression is exact, for
8644:, yields the operator equation
7773:
7772:
7701:
7648:
7644:
7534:
7294:
7132:
5164:. The arithmetic difference is
4669:, this results in a polynomial
1634:needs additional citations for
1451:is three times differentiable,
1420:
1308:
308:) and work by others including
29:Discrete analog of a derivative
13625:Arithmetico-geometric sequence
13317:Ordinary differential equation
12806:Levy, H.; Lessman, F. (1992).
12667:Calculus of Finite Differences
12491:
12480:
12469:
12431:
12417:
12397:
12151:
12127:
12118:
12100:
12091:
12073:
12064:
12052:
12040:
12022:
12013:
11995:
11986:
11962:
11950:
11938:
11878:
11854:
11845:
11821:
11812:
11788:
11779:
11755:
11739:
11727:
11694:
11676:
11667:
11655:
11643:
11625:
11609:
11597:
11564:
11546:
11537:
11525:
11513:
11495:
11479:
11467:
11436:
11418:
11409:
11391:
11375:
11363:
11335:
11317:
11308:
11290:
11274:
11262:
11106:is its coefficient vector. An
11069:
11054:
11014:
11008:
11005:
10999:
10941:
10929:
10920:
10914:
10901:
10895:
10861:
10855:
10846:
10834:
10821:
10815:
10748:
10733:
10431:
10422:
10366:
10357:
10267:
10243:
10204:{\displaystyle \ \Delta c=0\ }
10112:
10100:
10037:
10034:
10028:
9959:
9944:
9912:
9897:
9852:
9836:
9781:, the eigenfunction of
9736:
9729:
9703:
9696:
9670:
9663:
9556:
9549:
9531:
9524:
9330:
9323:
9237:
9225:
9066:
9047:
8895:
8876:
8601:
8595:
8561:
8555:
8540:
8528:
8509:
8503:
8453:
8447:
8438:
8432:
8130:Calculus of finite differences
8111:
8096:
8056:
8046:
7968:
7962:
7865:
7859:
7464:
7452:
7449:
7437:
7408:
7396:
7393:
7381:
7314:
7295:
7273:
7253:
7212:
7192:
7146:
7140:
6912:
6905:
6889:
6882:
6827:
6814:
6754:
6736:
6730:
6718:
6715:
6703:
6688:
6681:
6640:
6633:
6520:
6514:
6511:
6505:
6431:
6418:
6403:
6397:
6394:
6388:
6348:
6342:
6272:partial differential equations
6228:
6207:
6183:
6171:
6107:
6092:
5853:
5841:
5703:449 − 36(13) = 449 − 468 = −19
5693:341 − 36(10) = 341 − 360 = −19
5336:772 − 4(7) = 772 − 1372 = −600
5031:
5019:
5013:
4995:
4989:
4965:
4849:
4806:
4794:
4760:
4754:
4751:
4745:
4647:
4641:
4596:
4550:
4544:
4521:
4508:
4493:
4480:
4474:
4468:
4462:
4459:
4453:
4401:
4389:
4364:
4358:
4355:
4349:
4228:
4213:
4207:
4198:
4186:
4180:
4174:
4168:
4159:
4147:
4138:
4132:
4129:
4123:
4031:
4025:
4022:
4016:
3907:
3901:
3841:
3790:
3784:
3772:
3760:
3748:
3733:
3712:
3706:
3703:
3697:
3666:
3660:
3657:
3651:
3572:
3566:
3563:
3557:
3533:
3527:
3505:
3499:
3496:
3490:
3466:
3460:
3438:
3432:
3429:
3423:
3399:
3393:
3304:
3269:
3266:
3260:
3224:
3189:
3186:
3180:
3019:
3009:
2982:
2976:
2973:
2967:
2925:
2910:
2876:
2866:
2839:
2833:
2830:
2824:
2719:
2709:
2682:
2676:
2673:
2667:
2600:
2585:
2576:
2564:
2552:
2546:
2522:
2507:
2498:
2486:
2468:
2456:
2447:
2441:
2413:
2407:
2404:
2398:
2374:
2368:
2320:
2314:
2305:
2293:
2281:
2266:
2242:
2236:
2227:
2215:
2197:
2185:
2176:
2161:
2133:
2127:
2124:
2118:
2094:
2088:
2035:
2023:
2014:
2008:
1996:
1984:
1960:
1948:
1939:
1933:
1915:
1909:
1900:
1888:
1860:
1854:
1851:
1845:
1821:
1815:
1512:
1506:
1486:
1480:
1477:
1471:
1429:
1414:
1411:
1405:
1396:
1390:
1370:
1364:
1361:
1355:
1317:
1302:
1299:
1293:
1284:
1278:
1258:
1252:
1249:
1243:
1183:
1177:
1174:
1168:
1143:
1137:
1128:
1116:
1075:
1069:
1060:
1048:
1034:
1020:
1014:
954:
948:
945:
939:
915:
909:
906:
900:
876:
855:
846:
825:
816:
810:
807:
801:
758:
746:
743:
737:
721:
709:
700:
694:
685:
679:
676:
670:
599:
593:
584:
572:
563:
557:
554:
548:
532:
526:
523:
517:
481:
475:
466:
454:
445:
439:
436:
430:
390:
384:
323:
224:
218:
209:
197:
188:
182:
179:
173:
150:
144:
13:
1:
14070:Factorial and binomial topics
13448:Integro-differential equation
13322:Partial differential equation
12601:10.21468/SciPostPhys.10.1.007
12290:
11152:. Also one may make the step
10975:generalized finite difference
6796:of the generalization below.
6303:consists of the terms of the
5803:
5784:Finite difference coefficient
5683:233 − 36(7) = 233 − 252 = −19
5673:125 − 36(4) = 125 − 144 = −19
5323:109 − 4(4) = 109 − 256 = −147
1595:if it is calculated with the
302:
14075:Linear operators in calculus
13008:
12944:"Finite-difference calculus"
12899:10.1016/0304-3975(94)00281-M
12887:Theoretical Computer Science
10149:or antidifference operator.
8387:It also satisfies a special
6547:and for many (but not all)
4866:where the difference on the
4086:
3857:, expressed in the function
625:uses the function values at
507:is taken to be 1; that is,
399:{\displaystyle \Delta _{h},}
258:Certain recurrence relations
7:
13602:Generalized Stokes' theorem
13389:Integration by substitution
12949:Encyclopedia of Mathematics
12808:Finite Difference Equations
12629:, 2nd ed., Wiley, New York.
12625:and Morton, K.W., (1967).
12517:, Book III, Lemma V, Case 1
12230:
8155:. This operator amounts to
5160:This arrives at a constant
3631:. However, the combination
3335:Nörlund–Rice integral
639:, instead of the values at
10:
14101:
13131:(ε, δ)-definition of limit
12910:Richardson, C. H. (1954):
11150:weighted finite difference
8142:, which maps the function
6951:bosonic operator functions
6323:), first published in his
6311:; in essence, it is the
6251:
5787:
5781:
5663:17 − 36(1) = 17 − 36 = −19
5500:0 − (−17(1)) = 0 + 17 = 17
4846:This completes the proof.
4430:be a polynomial of degree
4293:be a polynomial of degree
4102:be a polynomial of degree
14024:Proof that 22/7 exceeds π
13961:
13939:
13865:
13813:Gottfried Wilhelm Leibniz
13783:
13760:e (mathematical constant)
13745:
13617:
13524:
13456:
13337:
13139:
13094:
13016:
12726:10.1142/S0217751X08040548
12680:– via Google books.
12664:Jordan, Charles (1965) .
12456:10.1017/S002026810002494X
12383:. Springer. p. 369.
12379:M Hanif Chaudhry (2007).
10008:Thus, for instance, the
9252:{\displaystyle \left==I.}
8818:, for sufficiently small
6551:. (It does not hold when
6276:finite difference methods
6248:In differential equations
5778:Arbitrarily sized kernels
4717:), it can be found that:
4706:pairwise differences for
3329:are sometimes called the
1597:central difference scheme
979:numerical differentiation
973:Relation with derivatives
413:is a function defined as
269:approximating derivatives
85:finite difference methods
13775:Stirling's approximation
13248:Implicit differentiation
13196:Rules of differentiation
12407:(Chelsea Pub Co, 2000)
8326:The difference operator
7744:the series of products,
6805:Chu–Vandermonde identity
6254:Finite difference method
1615:Higher-order differences
14009:Euler–Maclaurin formula
13914:trigonometric functions
13367:Constant of integration
12924:(Chapman and Hall/CRC)
12920:Mickens, R. E. (1991):
12784:10.3389/fphy.2013.00015
12512:Newton, Isaac, (1687).
8339:, as such it satisfies
156:{\displaystyle \Delta }
119:{\displaystyle \Delta }
97:boundary value problems
79:. The approximation of
14085:Non-Newtonian calculus
13978:Differential geometry
13823:Infinitesimal calculus
13526:Multivariable calculus
13474:Directional derivative
13280:Second derivative test
13258:Logarithmic derivative
13231:General Leibniz's rule
13126:Order of approximation
12670:. Chelsea Publishing.
12264:Gilbreath's conjecture
12175:
11906:
11238:in several variables.
11079:
11040:
10977:is usually defined as
10955:
10888:
10808:
10761:
10662:
10491:
10307:
10205:
10140:differential equations
10128:
10014:cardinal sine function
9984:
9769:
9581:
9491:
9253:
9137:
8989:
8789:
8636:Formally applying the
8624:
8204:
8121:
8045:
7928:is merely continuous.
7901:
7839:
7738:
7491:, Newton computes the
7479:
6928:
6859:
6761:
6664:
6592:
6572:
6530:
6463:
6374:
6238:
6133:
6059:
5963:
5926:
5892:
5769:
5613:
5438:
5247:
5038:
4838:
4654:
4411:
4266:
4060:
3973:
3825:up to a term of order
3808:
3619:
3317:
3234:
3105:
3008:
2935:
2865:
2792:
2708:
2623:
2343:
2058:
1603:is discrete. See also
1546:
1439:
1327:
1199:
1091:
964:
768:
609:
491:
400:
341:
234:
157:
120:
93:differential equations
14065:Mathematical analysis
13897:logarithmic functions
13892:exponential functions
13808:Generality of algebra
13686:Tests of convergence
13312:Differential equation
13296:Further applications
13285:Extreme value theorem
13275:First derivative test
13169:Differential operator
13141:Differential calculus
12914:(Van Nostrand (1954)
12176:
11907:
11197:partially ordered set
11177:modulus of continuity
11080:
11020:
10956:
10868:
10788:
10762:
10663:
10492:
10308:
10206:
10129:
9985:
9770:
9582:
9492:
9254:
9138:
8990:
8790:
8625:
8205:
8122:
8025:
7902:
7840:
7739:
7480:
6929:
6839:
6762:
6665:
6593:
6573:
6531:
6443:
6354:
6326:Principia Mathematica
6239:
6113:
6060:
5930:
5893:
5859:
5788:Further information:
5770:
5621:It can be found that
5614:
5439:
5248:
5039:
4839:
4655:
4412:
4267:
4061:
3974:
3809:
3620:
3318:
3235:
3116:binomial coefficients
3106:
2988:
2936:
2845:
2793:
2688:
2632:More generally, the
2624:
2351:Second order backward
2344:
2059:
1547:
1440:
1328:
1200:
1092:
965:
769:
610:
492:
401:
331:
250:differential equation
235:
158:
130:that maps a function
121:
13962:Miscellaneous topics
13902:hyperbolic functions
13887:irrational functions
13765:Exponential function
13618:Sequences and series
13384:Integration by parts
12874:Flajolet, Philippe;
12753:Frontiers in Physics
12500:/Mathologer (2021).
12427:. February 13, 2018.
12254:Finite volume method
11922:
11245:
10981:
10778:
10672:
10508:
10344:
10234:
10180:
10136:Difference equations
10019:
10010:Dirac delta function
9813:
9605:
9501:
9317:
9167:
9026:
8858:
8648:
8398:
8159:
7956:
7853:
7748:
7499:
6983:
6811:
6678:
6672:binomial coefficient
6602:
6591:{\displaystyle \pi }
6582:
6571:{\displaystyle \pi }
6562:
6539:which holds for any
6336:
6074:
5820:
5718:
5563:
5385:
5313:4 − 4(1) = 4 − 4 = 0
5179:
4962:
4721:
4440:
4325:
4110:
3998:
3895:
3868:, with real numbers
3635:
3356:
3339:asymptotic expansion
3244:
3164:
3114:These equations use
2949:
2806:
2649:
2357:
2077:
2071:Second order forward
1804:
1798:Second-order central
1643:improve this article
1605:Symmetric derivative
1455:
1339:
1227:
1107:
1003:
788:
657:
511:
417:
371:
356:finite differences.
167:
138:
110:
13949:List of derivatives
13785:History of calculus
13700:Cauchy condensation
13597:Exterior derivative
13554:Lagrange multiplier
13290:Maximum and minimum
13121:Limit of a sequence
13109:Limit of a function
13056:Graph of a function
13036:Continuous function
12965:D. Gleich (2005),
12831:Ames, W.F. (1977).
12775:2013FrP.....1...15C
12749:"Umbral Vade Mecum"
12718:2008IJMPA..23.2005Z
12592:2021ScPP...10....7K
12279:Time scale calculus
12243:Divided differences
11236:partial derivatives
11223:(1, −1, 0, 0, 0, …)
11148:, thus considering
11108:infinite difference
10998:
9999:generating function
9645:
9374:
9311:Pochhammer k-symbol
9216:
8972:
8941:
8140:difference operator
7493:divided differences
6283:thermal engineering
6206:
6170:
6091:
5978:
5840:
4744:
4348:
4015:
3885:(if any) marked as
3696:
3556:
3489:
3422:
2966:
2823:
2666:
2397:
2117:
1844:
1658:"Finite difference"
621:backward difference
284:L. M. Milne-Thomson
246:functional equation
242:difference equation
106:, commonly denoted
104:difference operator
76:difference quotient
14080:Numerical analysis
14055:Finite differences
13882:rational functions
13849:Method of Fluxions
13695:Alternating series
13592:Differential forms
13574:Partial derivative
13534:Divergence theorem
13416:Quadratic integral
13184:Leibniz's notation
13174:Mean value theorem
13159:Partial derivative
13104:Indeterminate form
12274:Summation by parts
12171:
11902:
11900:
11204:incidence algebras
11075:
10984:
10951:
10949:
10757:
10658:
10647:
10585:
10487:
10485:
10303:
10201:
10124:
10003:Pochhammer symbols
9980:
9765:
9628:
9577:
9487:
9357:
9249:
9199:
9133:
9109:
8985:
8958:
8955:
8927:
8924:
8830:finite stepsizes,
8816:analytic functions
8803:to its derivative
8785:
8620:
8200:
8117:
7989:
7897:
7835:
7734:
7475:
7473:
7469:
7416:
7366:
7292:
7231:
7128:
7100:
7074:
7056:
6958:Fibonacci sequence
6924:
6757:
6660:
6588:
6568:
6549:analytic functions
6526:
6260:numerical analysis
6234:
6186:
6156:
6077:
6055:
5964:
5823:
5790:Five-point stencil
5765:
5609:
5434:
5243:
5034:
4834:
4724:
4650:
4407:
4328:
4262:
4056:
4001:
3969:
3804:
3682:
3615:
3542:
3475:
3408:
3331:binomial transform
3313:
3299:
3230:
3219:
3101:
2952:
2931:
2809:
2788:
2652:
2619:
2383:
2339:
2103:
2054:
1830:
1542:
1435:
1323:
1195:
1087:
1041:
996:is defined by the
960:
874:
844:
780:central difference
764:
605:
487:
396:
363:forward difference
342:
265:numerical analysis
230:
153:
116:
18:Forward difference
14042:
14041:
13968:Complex calculus
13957:
13956:
13838:Law of Continuity
13770:Natural logarithm
13755:Bernoulli numbers
13746:Special functions
13705:Direct comparison
13569:Multiple integral
13443:Integral equation
13339:Integral calculus
13270:Stationary points
13244:Other techniques
13189:Newton's notation
13154:Second derivative
13046:Finite difference
12876:Sedgewick, Robert
12677:978-0-8284-0033-6
12390:978-0-387-68648-6
12381:Open-Channel Flow
12360:978-3-319-02099-0
12327:978-0-521-49789-3
12238:Discrete calculus
12166:
11893:
11708:
11578:
11448:
11347:
10946:
10787:
10756:
10752:
10690:
10529:
10513:
10482:
10353:
10302:
10293:
10278:
10263:
10251:
10239:
10200:
10185:
10120:
10116:
10083:
10062:
10024:
9976:
9970:
9923:
9884:
9863:
9834:
9818:
9764:
9754:
9721:
9688:
9627:
9610:
9573:
9548:
9522:
9506:
9483:
9470:
9356:
9350:
9322:
9190:
9129:
9108:
9073:
9000:asymptotic series
8981:
8954:
8923:
8781:
8726:
8688:
8616:
8606:
8576:
8566:
8551:
8545:
8524:
8514:
8484:
8474:
8468:
8458:
8443:
8428:
8403:
8264:identity operator
8196:
8085:
8017:
8013:
7974:
7947:difference series
7932:Carlson's theorem
7909:In analysis with
7642:
7415:
7365:
7291:
7230:
7093:
7067:
7049:
6874:
6769:falling factorial
6658:
6619:
6486:
6415:
6148:
5809:For all positive
5708:
5707:
5557:
5556:
5379:
5378:
5158:
5157:
4934:
4933:
4633:
4594:
4542:
3883:lower order terms
3802:
3719:
3680:
3586:
3519:
3452:
3391:
3309:
3303:
3298:
3293:
3287:
3274:
3249:
3229:
3223:
3218:
3213:
3207:
3194:
3169:
3143:Pascal's triangle
3077:
3042:
2899:
2748:
2614:
2535:
2529:
2475:
2427:
2334:
2255:
2249:
2204:
2147:
2049:
1973:
1967:
1922:
1874:
1719:
1718:
1711:
1693:
1493:
1424:
1377:
1312:
1265:
1190:
1150:
1082:
1026:
873:
843:
34:finite difference
16:(Redirected from
14092:
13972:Contour integral
13870:
13869:
13720:Limit comparison
13629:Types of series
13588:Advanced topics
13579:Surface integral
13423:Trapezoidal rule
13362:Basic properties
13357:Riemann integral
13305:Taylor's theorem
13031:Concave function
13026:Binomial theorem
13003:
12996:
12989:
12980:
12979:
12957:
12903:
12902:
12893:(1–2): 101–124.
12884:
12871:
12865:
12864:
12857:Hildebrand, F.B.
12853:
12847:
12846:
12828:
12822:
12821:
12803:
12797:
12796:
12786:
12768:
12744:
12738:
12737:
12711:
12688:
12682:
12681:
12661:
12655:
12653:
12650:Internet Archive
12636:
12630:
12620:
12614:
12613:
12603:
12585:
12561:
12555:
12554:
12552:
12534:
12525:
12519:
12510:
12504:
12495:
12489:
12484:
12478:
12473:
12467:
12466:
12464:
12462:
12435:
12429:
12428:
12421:
12415:
12401:
12395:
12394:
12376:
12365:
12364:
12343:
12332:
12331:
12315:
12305:
12269:Sheffer sequence
12226:
12203:
12180:
12178:
12177:
12172:
12167:
12165:
12154:
11957:
11937:
11936:
11917:
11911:
11909:
11908:
11903:
11901:
11894:
11892:
11881:
11750:
11726:
11725:
11709:
11707:
11706:
11697:
11620:
11596:
11595:
11579:
11577:
11576:
11567:
11490:
11466:
11465:
11449:
11447:
11439:
11386:
11362:
11361:
11348:
11346:
11338:
11285:
11261:
11260:
11224:
11221:is the sequence
11220:
11216:
11193:Möbius inversion
11187:
11174:
11159:
11156:depend on point
11155:
11147:
11126:
11123:depend on point
11122:
11105:
11084:
11082:
11081:
11076:
11050:
11049:
11039:
11034:
10997:
10992:
10964:See references.
10960:
10958:
10957:
10952:
10950:
10944:
10887:
10882:
10807:
10802:
10785:
10766:
10764:
10763:
10758:
10754:
10753:
10751:
10725:
10700:
10695:
10691:
10683:
10667:
10665:
10664:
10659:
10657:
10653:
10652:
10651:
10599:
10595:
10591:
10590:
10589:
10534:
10530:
10522:
10511:
10496:
10494:
10493:
10488:
10486:
10480:
10351:
10331:
10329:
10323:
10319:
10312:
10310:
10309:
10304:
10300:
10291:
10276:
10261:
10249:
10237:
10225:
10221:
10210:
10208:
10207:
10202:
10198:
10183:
10171:
10133:
10131:
10130:
10125:
10118:
10117:
10115:
10095:
10094:
10090:
10089:
10085:
10084:
10076:
10063:
10055:
10041:
10022:
9989:
9987:
9986:
9981:
9974:
9973:
9972:
9971:
9963:
9926:
9925:
9924:
9916:
9885:
9880:
9879:
9870:
9865:
9864:
9856:
9835:
9830:
9829:
9820:
9816:
9805:
9804:
9802:
9801:
9796:
9793:
9774:
9772:
9771:
9766:
9762:
9755:
9753:
9745:
9744:
9743:
9727:
9722:
9720:
9712:
9711:
9710:
9694:
9689:
9687:
9679:
9678:
9677:
9661:
9650:
9646:
9644:
9636:
9625:
9608:
9597:
9586:
9584:
9583:
9578:
9571:
9570:
9569:
9546:
9539:
9538:
9523:
9518:
9517:
9508:
9504:
9496:
9494:
9493:
9488:
9481:
9480:
9479:
9468:
9467:
9463:
9442:
9441:
9432:
9428:
9410:
9406:
9385:
9384:
9379:
9375:
9373:
9365:
9354:
9348:
9338:
9337:
9320:
9308:
9302:
9300:
9298:
9297:
9272:
9258:
9256:
9255:
9250:
9221:
9217:
9215:
9207:
9191:
9186:
9185:
9176:
9159:
9142:
9140:
9139:
9134:
9127:
9126:
9122:
9121:
9120:
9110:
9101:
9074:
9071:
9065:
9064:
9012:
8994:
8992:
8991:
8986:
8979:
8971:
8966:
8956:
8947:
8940:
8935:
8925:
8916:
8910:
8909:
8894:
8893:
8853:
8851:
8836:
8834:
8821:
8813:
8811:
8802:
8798:
8794:
8792:
8791:
8786:
8779:
8769:
8768:
8747:
8746:
8737:
8736:
8727:
8725:
8714:
8709:
8708:
8699:
8698:
8689:
8687:
8676:
8662:
8661:
8656:
8643:
8640:with respect to
8629:
8627:
8626:
8621:
8614:
8613:
8612:
8604:
8588:
8587:
8582:
8574:
8573:
8572:
8564:
8549:
8543:
8522:
8521:
8520:
8512:
8496:
8495:
8490:
8482:
8481:
8480:
8472:
8466:
8465:
8464:
8456:
8441:
8426:
8425:
8424:
8415:
8414:
8409:
8401:
8384:
8382:
8334:
8323:
8321:
8319:
8318:
8304:
8302:
8300:
8299:
8283:
8282:
8261:
8257:
8255:
8227:
8219:
8209:
8207:
8206:
8201:
8194:
8184:
8183:
8171:
8170:
8154:
8145:
8126:
8124:
8123:
8118:
8092:
8091:
8090:
8077:
8070:
8069:
8044:
8039:
8024:
8023:
8022:
8009:
7998:
7996:
7988:
7927:
7923:
7918:Mahler's theorem
7913:
7906:
7904:
7903:
7898:
7893:
7844:
7842:
7841:
7836:
7831:
7827:
7826:
7825:
7802:
7801:
7789:
7788:
7762:
7761:
7760:
7743:
7741:
7740:
7735:
7733:
7732:
7714:
7713:
7697:
7693:
7686:
7643:
7641:
7640:
7639:
7627:
7626:
7610:
7609:
7608:
7584:
7583:
7555:
7550:
7549:
7530:
7529:
7517:
7516:
7490:
7484:
7482:
7481:
7476:
7474:
7470:
7424:
7421:
7417:
7411:
7379:
7367:
7361:
7350:
7323:
7320:
7307:
7306:
7293:
7290:
7282:
7281:
7280:
7271:
7270:
7251:
7245:
7244:
7232:
7229:
7221:
7220:
7219:
7210:
7209:
7190:
7184:
7183:
7165:
7164:
7129:
7126:
7125:
7112:
7106:
7105:
7101:
7090:
7077:
7075:
7064:
7063:
7060:
7059:
7057:
7038:
7037:
7026:
7025:
7014:
7013:
6978:
6965:
6936:binomial theorem
6933:
6931:
6930:
6925:
6920:
6919:
6903:
6902:
6881:
6880:
6879:
6866:
6858:
6853:
6835:
6834:
6801:Taylor's theorem
6795:
6784:
6766:
6764:
6763:
6758:
6696:
6695:
6669:
6667:
6666:
6661:
6659:
6657:
6649:
6648:
6647:
6631:
6626:
6625:
6624:
6611:
6597:
6595:
6594:
6589:
6577:
6575:
6574:
6569:
6557:exponential type
6554:
6546:
6535:
6533:
6532:
6527:
6504:
6503:
6493:
6492:
6491:
6482:
6470:
6462:
6457:
6439:
6438:
6416:
6414:
6406:
6387:
6386:
6376:
6373:
6368:
6262:, especially in
6243:
6241:
6240:
6235:
6205:
6194:
6169:
6164:
6155:
6154:
6153:
6140:
6132:
6127:
6090:
6085:
6064:
6062:
6061:
6056:
6051:
6047:
6043:
6042:
6021:
6020:
6005:
6004:
5977:
5972:
5962:
5951:
5944:
5943:
5925:
5914:
5907:
5906:
5891:
5880:
5873:
5872:
5839:
5834:
5816:
5812:
5797:Taylor expansion
5774:
5772:
5771:
5766:
5749:
5748:
5733:
5732:
5713:
5704:
5694:
5684:
5674:
5664:
5654:
5647:
5640:
5639:
5636:
5627:
5618:
5616:
5615:
5610:
5587:
5586:
5550:
5537:
5524:
5511:
5501:
5491:
5483:
5476:
5469:
5468:
5462:
5455:
5451:
5443:
5441:
5440:
5435:
5412:
5411:
5369:
5353:
5337:
5324:
5314:
5304:
5296:
5288:
5283:
5278:
5277:
5271:
5264:
5260:
5252:
5250:
5249:
5244:
5203:
5202:
5174:
5170:
5163:
5087:
5079:
5071:
5063:
5056:
5049:
5048:
5043:
5041:
5040:
5035:
4957:
4953:
4941:
4880:
4879:
4876:
4865:
4843:
4841:
4840:
4835:
4827:
4826:
4793:
4792:
4743:
4732:
4716:
4705:
4699:
4692:
4686:
4679:
4668:
4659:
4657:
4656:
4651:
4634:
4631:
4626:
4625:
4595:
4592:
4587:
4586:
4565:
4564:
4543:
4540:
4535:
4534:
4501:
4500:
4452:
4451:
4435:
4429:
4416:
4414:
4413:
4408:
4388:
4387:
4347:
4336:
4320:
4313:
4306:
4299:
4292:
4271:
4269:
4268:
4263:
4255:
4254:
4122:
4121:
4105:
4101:
4077:
4073:
4065:
4063:
4062:
4057:
4049:
4048:
4014:
4009:
3993:
3986:
3978:
3976:
3975:
3970:
3947:
3946:
3925:
3924:
3890:
3880:
3874:
3867:
3856:
3830:
3824:
3813:
3811:
3810:
3805:
3803:
3801:
3793:
3728:
3720:
3715:
3695:
3690:
3681:
3673:
3650:
3649:
3639:
3630:
3624:
3622:
3621:
3616:
3611:
3607:
3606:
3587:
3585:
3584:
3575:
3555:
3550:
3540:
3520:
3518:
3517:
3508:
3488:
3483:
3473:
3453:
3451:
3450:
3441:
3421:
3416:
3406:
3392:
3390:
3389:
3388:
3375:
3371:
3370:
3360:
3348:
3322:
3320:
3319:
3314:
3307:
3301:
3300:
3294:
3291:
3285:
3283:
3272:
3259:
3258:
3247:
3239:
3237:
3236:
3231:
3227:
3221:
3220:
3214:
3211:
3205:
3203:
3192:
3179:
3178:
3167:
3159:
3155:
3148:
3140:
3134:
3133:
3110:
3108:
3107:
3102:
3097:
3093:
3089:
3085:
3078:
3070:
3049:
3048:
3047:
3034:
3027:
3026:
3007:
3002:
2965:
2960:
2940:
2938:
2937:
2932:
2906:
2905:
2904:
2891:
2884:
2883:
2864:
2859:
2822:
2817:
2797:
2795:
2794:
2789:
2784:
2783:
2765:
2764:
2755:
2754:
2753:
2740:
2733:
2732:
2707:
2702:
2665:
2660:
2636:
2628:
2626:
2625:
2620:
2615:
2613:
2612:
2603:
2541:
2536:
2531:
2530:
2525:
2481:
2476:
2471:
2436:
2433:
2428:
2426:
2425:
2416:
2396:
2391:
2381:
2367:
2348:
2346:
2345:
2340:
2335:
2333:
2332:
2323:
2261:
2256:
2251:
2250:
2245:
2210:
2205:
2200:
2156:
2153:
2148:
2146:
2145:
2136:
2116:
2111:
2101:
2087:
2063:
2061:
2060:
2055:
2050:
2048:
2047:
2038:
1979:
1974:
1969:
1968:
1963:
1928:
1923:
1918:
1883:
1880:
1875:
1873:
1872:
1863:
1843:
1838:
1828:
1814:
1793:
1789:
1785:
1778:
1776:
1774:
1773:
1770:
1767:
1749:
1747:
1745:
1744:
1741:
1738:
1714:
1707:
1703:
1700:
1694:
1692:
1651:
1627:
1619:
1602:
1594:
1583:
1579:
1568:
1564:
1551:
1549:
1548:
1543:
1538:
1534:
1533:
1505:
1494:
1489:
1470:
1469:
1459:
1450:
1444:
1442:
1441:
1436:
1425:
1422:
1389:
1378:
1373:
1354:
1353:
1343:
1332:
1330:
1329:
1324:
1313:
1310:
1277:
1266:
1261:
1242:
1241:
1231:
1222:
1219:. Assuming that
1217:Taylor's theorem
1214:
1210:
1204:
1202:
1201:
1196:
1191:
1186:
1167:
1166:
1156:
1151:
1146:
1111:
1102:
1096:
1094:
1093:
1088:
1083:
1078:
1043:
1040:
1013:
995:
991:
969:
967:
966:
961:
938:
937:
933:
899:
898:
894:
875:
866:
845:
836:
800:
799:
782:
781:
773:
771:
770:
765:
736:
735:
669:
668:
652:
648:
638:
628:
623:
622:
614:
612:
611:
606:
547:
546:
506:
502:
496:
494:
493:
488:
429:
428:
412:
405:
403:
402:
397:
383:
382:
365:
364:
307:
304:
296:
239:
237:
236:
231:
162:
160:
159:
154:
134:to the function
133:
125:
123:
122:
117:
72:
62:
21:
14100:
14099:
14095:
14094:
14093:
14091:
14090:
14089:
14045:
14044:
14043:
14038:
14034:Steinmetz solid
14019:Integration Bee
13953:
13935:
13861:
13803:Colin Maclaurin
13779:
13747:
13741:
13613:
13607:Tensor calculus
13584:Volume integral
13520:
13495:Basic theorems
13458:Vector calculus
13452:
13333:
13300:Newton's method
13135:
13114:One-sided limit
13090:
13071:Rolle's theorem
13061:Linear function
13012:
13007:
12942:
12939:
12934:
12906:
12882:
12872:
12868:
12854:
12850:
12843:
12829:
12825:
12818:
12804:
12800:
12745:
12741:
12702:(13): 200–214.
12689:
12685:
12678:
12662:
12658:
12637:
12633:
12621:
12617:
12570:SciPost Physics
12562:
12558:
12532:
12526:
12522:
12511:
12507:
12498:Burkard Polster
12496:
12492:
12485:
12481:
12474:
12470:
12460:
12458:
12436:
12432:
12423:
12422:
12418:
12402:
12398:
12391:
12377:
12368:
12361:
12344:
12335:
12328:
12306:
12297:
12293:
12288:
12233:
12205:
12182:
12155:
11958:
11956:
11929:
11925:
11923:
11920:
11919:
11915:
11899:
11898:
11882:
11751:
11749:
11742:
11718:
11714:
11711:
11710:
11702:
11698:
11621:
11619:
11612:
11588:
11584:
11581:
11580:
11572:
11568:
11491:
11489:
11482:
11458:
11454:
11451:
11450:
11440:
11387:
11385:
11378:
11357:
11353:
11350:
11349:
11339:
11286:
11284:
11277:
11256:
11252:
11248:
11246:
11243:
11242:
11232:
11222:
11218:
11214:
11212:Möbius function
11183:
11161:
11157:
11153:
11140:
11133:
11128:
11124:
11120:
11115:
11112:infinite series
11102:
11096:
11086:
11045:
11041:
11035:
11024:
10993:
10988:
10982:
10979:
10978:
10970:
10968:Generalizations
10948:
10947:
10904:
10883:
10872:
10865:
10864:
10824:
10803:
10792:
10781:
10779:
10776:
10775:
10771:Summation rules
10726:
10701:
10699:
10682:
10678:
10673:
10670:
10669:
10646:
10645:
10640:
10634:
10633:
10625:
10615:
10614:
10604:
10600:
10584:
10583:
10578:
10572:
10571:
10563:
10550:
10549:
10545:
10541:
10538:
10521:
10517:
10509:
10506:
10505:
10484:
10483:
10434:
10416:
10415:
10369:
10347:
10345:
10342:
10341:
10327:
10325:
10321:
10317:
10235:
10232:
10231:
10223:
10219:
10181:
10178:
10177:
10169:
10155:
10096:
10075:
10068:
10064:
10054:
10053:
10049:
10042:
10040:
10020:
10017:
10016:
9962:
9937:
9933:
9915:
9890:
9886:
9875:
9871:
9869:
9855:
9851:
9825:
9821:
9819:
9814:
9811:
9810:
9797:
9794:
9792:
9786:
9785:
9783:
9782:
9779:continuum limit
9746:
9739:
9735:
9728:
9726:
9713:
9706:
9702:
9695:
9693:
9680:
9673:
9669:
9662:
9660:
9637:
9632:
9621:
9617:
9606:
9603:
9602:
9588:
9559:
9555:
9534:
9530:
9513:
9509:
9507:
9502:
9499:
9498:
9475:
9474:
9453:
9449:
9437:
9436:
9415:
9411:
9396:
9392:
9380:
9366:
9361:
9347:
9343:
9342:
9333:
9329:
9318:
9315:
9314:
9306:
9296:
9291:
9290:
9289:
9280:
9278:
9263:
9260:
9208:
9203:
9181:
9177:
9175:
9174:
9170:
9168:
9165:
9164:
9154:
9147:umbral calculus
9116:
9112:
9099:
9098:
9094:
9072: and
9070:
9060:
9056:
9027:
9024:
9023:
9003:
8967:
8962:
8945:
8936:
8931:
8914:
8905:
8901:
8889:
8885:
8859:
8856:
8855:
8846:
8840:
8838:
8832:
8831:
8819:
8806:
8804:
8800:
8796:
8761:
8757:
8742:
8738:
8732:
8728:
8718:
8713:
8704:
8700:
8694:
8690:
8680:
8675:
8657:
8652:
8651:
8649:
8646:
8645:
8641:
8608:
8607:
8583:
8578:
8577:
8568:
8567:
8516:
8515:
8491:
8486:
8485:
8476:
8475:
8460:
8459:
8420:
8419:
8410:
8405:
8404:
8399:
8396:
8395:
8376:
8362:
8348:
8342:
8340:
8337:linear operator
8333:
8327:
8317:
8312:
8311:
8310:
8308:
8306:
8298:
8293:
8292:
8291:
8289:
8281:
8276:
8275:
8274:
8272:
8270:
8260: I
8259:
8237:
8231:
8229:
8225:
8217:
8211:
8179:
8175:
8166:
8162:
8160:
8157:
8156:
8153:
8147:
8143:
8132:
8086:
8073:
8072:
8071:
8059:
8055:
8040:
8029:
8018:
7999:
7992:
7991:
7990:
7978:
7957:
7954:
7953:
7939:Stirling series
7925:
7921:
7911:
7883:
7854:
7851:
7850:
7821:
7817:
7810:
7806:
7797:
7793:
7778:
7774:
7756:
7752:
7751:
7749:
7746:
7745:
7722:
7718:
7709:
7705:
7676:
7653:
7649:
7635:
7631:
7616:
7612:
7611:
7592:
7588:
7561:
7557:
7556:
7554:
7539:
7535:
7525:
7521:
7506:
7502:
7500:
7497:
7496:
7488:
7472:
7471:
7468:
7467:
7422:
7419:
7418:
7380:
7377:
7351:
7348:
7321:
7318:
7317:
7302:
7298:
7283:
7276:
7272:
7266:
7262:
7252:
7249:
7240:
7236:
7222:
7215:
7211:
7205:
7201:
7191:
7188:
7179:
7175:
7160:
7156:
7149:
7133:
7130:
7127:
7124:
7119:
7113:
7111:
7103:
7102:
7092:
7089:
7084:
7078:
7076:
7066:
7061:
7058:
7048:
7046:
7040:
7039:
7033:
7029:
7027:
7021:
7017:
7015:
7009:
7005:
6997:
6990:
6986:
6984:
6981:
6980:
6977:
6971:
6966:One can find a
6960:
6940:umbral calculus
6915:
6911:
6892:
6888:
6875:
6862:
6861:
6860:
6854:
6843:
6830:
6826:
6812:
6809:
6808:
6786:
6783:
6775:
6691:
6687:
6679:
6676:
6675:
6650:
6643:
6639:
6632:
6630:
6620:
6607:
6606:
6605:
6603:
6600:
6599:
6583:
6580:
6579:
6563:
6560:
6559:
6552:
6544:
6537:
6499:
6495:
6487:
6472:
6466:
6465:
6464:
6458:
6447:
6434:
6430:
6407:
6382:
6378:
6377:
6375:
6369:
6358:
6337:
6334:
6333:
6295:
6293:Newton's series
6287:fluid mechanics
6256:
6250:
6195:
6190:
6165:
6160:
6149:
6136:
6135:
6134:
6128:
6117:
6086:
6081:
6075:
6072:
6071:
6038:
6034:
6016:
6012:
6000:
5996:
5983:
5979:
5973:
5968:
5952:
5939:
5935:
5934:
5915:
5902:
5898:
5897:
5881:
5868:
5864:
5863:
5835:
5827:
5821:
5818:
5817:
5814:
5810:
5806:
5792:
5786:
5780:
5744:
5740:
5728:
5724:
5719:
5716:
5715:
5711:
5702:
5692:
5682:
5672:
5662:
5650:
5643:
5629:
5622:
5582:
5578:
5564:
5561:
5560:
5548:
5535:
5522:
5509:
5499:
5486:
5479:
5472:
5457:
5453:
5447:
5407:
5403:
5386:
5383:
5382:
5367:
5351:
5335:
5322:
5312:
5299:
5291:
5286:
5281:
5266:
5262:
5256:
5198:
5194:
5180:
5177:
5176:
5172:
5165:
5161:
5082:
5074:
5066:
5059:
5052:
4963:
4960:
4959:
4955:
4943:
4937:
4871:
4855:
4852:
4822:
4818:
4782:
4778:
4733:
4728:
4722:
4719:
4718:
4707:
4701:
4694:
4688:
4681:
4670:
4663:
4630:
4615:
4611:
4591:
4576:
4572:
4560:
4556:
4539:
4524:
4520:
4496:
4492:
4447:
4443:
4441:
4438:
4437:
4431:
4420:
4377:
4373:
4337:
4332:
4326:
4323:
4322:
4315:
4308:
4301:
4294:
4283:
4280:
4250:
4246:
4117:
4113:
4111:
4108:
4107:
4103:
4092:
4089:
4084:
4082:Inductive proof
4075:
4069:
4044:
4040:
4010:
4005:
3999:
3996:
3995:
3988:
3982:
3936:
3932:
3920:
3916:
3896:
3893:
3892:
3886:
3876:
3869:
3858:
3851:
3844:
3826:
3815:
3794:
3729:
3727:
3691:
3686:
3672:
3645:
3641:
3640:
3638:
3636:
3633:
3632:
3628:
3602:
3598:
3594:
3580:
3576:
3551:
3546:
3541:
3539:
3513:
3509:
3484:
3479:
3474:
3472:
3446:
3442:
3417:
3412:
3407:
3405:
3384:
3380:
3376:
3366:
3362:
3361:
3359:
3357:
3354:
3353:
3346:
3284:
3281:
3254:
3250:
3245:
3242:
3241:
3204:
3201:
3174:
3170:
3165:
3162:
3161:
3157:
3153:
3146:
3141:. Each row of
3139:
3138:
3132:
3127:
3126:
3125:
3124:
3123:
3119:
3069:
3068:
3064:
3057:
3053:
3043:
3030:
3029:
3028:
3022:
3018:
3003:
2992:
2961:
2956:
2950:
2947:
2946:
2900:
2887:
2886:
2885:
2879:
2875:
2860:
2849:
2818:
2813:
2807:
2804:
2803:
2779:
2778:
2760:
2759:
2749:
2736:
2735:
2734:
2722:
2718:
2703:
2692:
2661:
2656:
2650:
2647:
2646:
2634:
2608:
2604:
2542:
2540:
2482:
2480:
2437:
2435:
2434:
2432:
2421:
2417:
2392:
2387:
2382:
2380:
2360:
2358:
2355:
2354:
2328:
2324:
2262:
2260:
2211:
2209:
2157:
2155:
2154:
2152:
2141:
2137:
2112:
2107:
2102:
2100:
2080:
2078:
2075:
2074:
2043:
2039:
1980:
1978:
1929:
1927:
1884:
1882:
1881:
1879:
1868:
1864:
1839:
1834:
1829:
1827:
1807:
1805:
1802:
1801:
1791:
1787:
1780:
1771:
1768:
1763:
1762:
1760:
1751:
1742:
1739:
1734:
1733:
1731:
1722:
1715:
1704:
1698:
1695:
1652:
1650:
1640:
1628:
1617:
1600:
1585:
1581:
1570:
1566:
1555:
1529:
1525:
1521:
1498:
1465:
1461:
1460:
1458:
1456:
1453:
1452:
1448:
1421:
1382:
1349:
1345:
1344:
1342:
1340:
1337:
1336:
1309:
1270:
1237:
1233:
1232:
1230:
1228:
1225:
1224:
1220:
1212:
1208:
1162:
1158:
1157:
1155:
1112:
1110:
1108:
1105:
1104:
1100:
1044:
1042:
1030:
1006:
1004:
1001:
1000:
993:
989:
975:
929:
925:
921:
890:
886:
882:
864:
834:
795:
791:
789:
786:
785:
779:
778:
731:
727:
664:
660:
658:
655:
654:
650:
640:
630:
626:
620:
619:
542:
538:
512:
509:
508:
504:
500:
424:
420:
418:
415:
414:
410:
378:
374:
372:
369:
368:
362:
361:
326:
305:
301:'s algorithms (
290:
168:
165:
164:
139:
136:
135:
131:
111:
108:
107:
64:
37:
30:
23:
22:
15:
12:
11:
5:
14098:
14088:
14087:
14082:
14077:
14072:
14067:
14062:
14057:
14040:
14039:
14037:
14036:
14031:
14026:
14021:
14016:
14014:Gabriel's horn
14011:
14006:
14005:
14004:
13999:
13994:
13989:
13984:
13976:
13975:
13974:
13965:
13963:
13959:
13958:
13955:
13954:
13952:
13951:
13946:
13944:List of limits
13940:
13937:
13936:
13934:
13933:
13932:
13931:
13926:
13921:
13911:
13910:
13909:
13899:
13894:
13889:
13884:
13878:
13876:
13867:
13863:
13862:
13860:
13859:
13852:
13845:
13843:Leonhard Euler
13840:
13835:
13830:
13825:
13820:
13815:
13810:
13805:
13800:
13795:
13789:
13787:
13781:
13780:
13778:
13777:
13772:
13767:
13762:
13757:
13751:
13749:
13743:
13742:
13740:
13739:
13738:
13737:
13732:
13727:
13722:
13717:
13712:
13707:
13702:
13697:
13692:
13684:
13683:
13682:
13677:
13676:
13675:
13670:
13660:
13655:
13650:
13645:
13640:
13635:
13627:
13621:
13619:
13615:
13614:
13612:
13611:
13610:
13609:
13604:
13599:
13594:
13586:
13581:
13576:
13571:
13566:
13561:
13556:
13551:
13546:
13544:Hessian matrix
13541:
13536:
13530:
13528:
13522:
13521:
13519:
13518:
13517:
13516:
13511:
13506:
13501:
13499:Line integrals
13493:
13492:
13491:
13486:
13481:
13476:
13471:
13462:
13460:
13454:
13453:
13451:
13450:
13445:
13440:
13439:
13438:
13433:
13425:
13420:
13419:
13418:
13408:
13407:
13406:
13401:
13396:
13386:
13381:
13380:
13379:
13369:
13364:
13359:
13354:
13349:
13347:Antiderivative
13343:
13341:
13335:
13334:
13332:
13331:
13330:
13329:
13324:
13319:
13309:
13308:
13307:
13302:
13294:
13293:
13292:
13287:
13282:
13277:
13267:
13266:
13265:
13260:
13255:
13250:
13242:
13241:
13240:
13235:
13234:
13233:
13223:
13218:
13213:
13208:
13203:
13193:
13192:
13191:
13186:
13176:
13171:
13166:
13161:
13156:
13151:
13145:
13143:
13137:
13136:
13134:
13133:
13128:
13123:
13118:
13117:
13116:
13106:
13100:
13098:
13092:
13091:
13089:
13088:
13083:
13078:
13073:
13068:
13063:
13058:
13053:
13048:
13043:
13038:
13033:
13028:
13022:
13020:
13014:
13013:
13006:
13005:
12998:
12991:
12983:
12977:
12976:
12971:
12963:
12958:
12938:
12937:External links
12935:
12933:
12932:
12930:978-0442001360
12918:
12907:
12905:
12904:
12866:
12848:
12841:
12823:
12816:
12798:
12739:
12683:
12676:
12656:
12631:
12623:Richtmeyer, D.
12615:
12556:
12520:
12505:
12490:
12479:
12468:
12450:(2): 235–241.
12430:
12416:
12413:978-0821821077
12396:
12389:
12366:
12359:
12333:
12326:
12294:
12292:
12289:
12287:
12286:
12281:
12276:
12271:
12266:
12261:
12256:
12251:
12245:
12240:
12234:
12232:
12229:
12170:
12164:
12161:
12158:
12153:
12150:
12147:
12144:
12141:
12138:
12135:
12132:
12129:
12126:
12123:
12120:
12117:
12114:
12111:
12108:
12105:
12102:
12099:
12096:
12093:
12090:
12087:
12084:
12081:
12078:
12075:
12072:
12069:
12066:
12063:
12060:
12057:
12054:
12051:
12048:
12045:
12042:
12039:
12036:
12033:
12030:
12027:
12024:
12021:
12018:
12015:
12012:
12009:
12006:
12003:
12000:
11997:
11994:
11991:
11988:
11985:
11982:
11979:
11976:
11973:
11970:
11967:
11964:
11961:
11955:
11952:
11949:
11946:
11943:
11940:
11935:
11932:
11928:
11897:
11891:
11888:
11885:
11880:
11877:
11874:
11871:
11868:
11865:
11862:
11859:
11856:
11853:
11850:
11847:
11844:
11841:
11838:
11835:
11832:
11829:
11826:
11823:
11820:
11817:
11814:
11811:
11808:
11805:
11802:
11799:
11796:
11793:
11790:
11787:
11784:
11781:
11778:
11775:
11772:
11769:
11766:
11763:
11760:
11757:
11754:
11748:
11745:
11743:
11741:
11738:
11735:
11732:
11729:
11724:
11721:
11717:
11713:
11712:
11705:
11701:
11696:
11693:
11690:
11687:
11684:
11681:
11678:
11675:
11672:
11669:
11666:
11663:
11660:
11657:
11654:
11651:
11648:
11645:
11642:
11639:
11636:
11633:
11630:
11627:
11624:
11618:
11615:
11613:
11611:
11608:
11605:
11602:
11599:
11594:
11591:
11587:
11583:
11582:
11575:
11571:
11566:
11563:
11560:
11557:
11554:
11551:
11548:
11545:
11542:
11539:
11536:
11533:
11530:
11527:
11524:
11521:
11518:
11515:
11512:
11509:
11506:
11503:
11500:
11497:
11494:
11488:
11485:
11483:
11481:
11478:
11475:
11472:
11469:
11464:
11461:
11457:
11453:
11452:
11446:
11443:
11438:
11435:
11432:
11429:
11426:
11423:
11420:
11417:
11414:
11411:
11408:
11405:
11402:
11399:
11396:
11393:
11390:
11384:
11381:
11379:
11377:
11374:
11371:
11368:
11365:
11360:
11356:
11352:
11351:
11345:
11342:
11337:
11334:
11331:
11328:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11304:
11301:
11298:
11295:
11292:
11289:
11283:
11280:
11278:
11276:
11273:
11270:
11267:
11264:
11259:
11255:
11251:
11250:
11231:
11228:
11227:
11226:
11200:
11189:
11180:
11138:
11131:
11118:
11100:
11094:
11074:
11071:
11068:
11065:
11062:
11059:
11056:
11053:
11048:
11044:
11038:
11033:
11030:
11027:
11023:
11019:
11016:
11013:
11010:
11007:
11004:
11001:
10996:
10991:
10987:
10969:
10966:
10962:
10961:
10943:
10940:
10937:
10934:
10931:
10928:
10925:
10922:
10919:
10916:
10913:
10910:
10907:
10905:
10903:
10900:
10897:
10894:
10891:
10886:
10881:
10878:
10875:
10871:
10867:
10866:
10863:
10860:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10836:
10833:
10830:
10827:
10825:
10823:
10820:
10817:
10814:
10811:
10806:
10801:
10798:
10795:
10791:
10784:
10783:
10767:
10750:
10747:
10744:
10741:
10738:
10735:
10732:
10729:
10724:
10721:
10717:
10714:
10711:
10708:
10704:
10698:
10694:
10689:
10686:
10681:
10677:
10656:
10650:
10644:
10641:
10639:
10636:
10635:
10632:
10629:
10626:
10624:
10621:
10620:
10618:
10613:
10610:
10607:
10603:
10598:
10594:
10588:
10582:
10579:
10577:
10574:
10573:
10570:
10567:
10564:
10562:
10559:
10556:
10555:
10553:
10548:
10544:
10540:
10537:
10533:
10528:
10525:
10520:
10516:
10497:
10479:
10476:
10472:
10469:
10466:
10463:
10460:
10456:
10453:
10450:
10447:
10443:
10440:
10437:
10435:
10433:
10430:
10427:
10424:
10421:
10418:
10417:
10414:
10411:
10407:
10404:
10401:
10398:
10395:
10391:
10388:
10385:
10382:
10378:
10375:
10372:
10370:
10368:
10365:
10362:
10359:
10356:
10350:
10349:
10314:
10313:
10299:
10296:
10290:
10287:
10284:
10281:
10275:
10272:
10269:
10266:
10260:
10257:
10254:
10248:
10245:
10242:
10211:
10197:
10194:
10191:
10188:
10154:
10151:
10147:indefinite sum
10134:and so forth.
10123:
10114:
10111:
10108:
10105:
10102:
10099:
10093:
10088:
10082:
10079:
10074:
10071:
10067:
10061:
10058:
10052:
10048:
10045:
10039:
10036:
10033:
10030:
10027:
9991:
9990:
9979:
9969:
9966:
9961:
9958:
9955:
9952:
9949:
9946:
9943:
9940:
9936:
9932:
9929:
9922:
9919:
9914:
9911:
9908:
9905:
9902:
9899:
9896:
9893:
9889:
9883:
9878:
9874:
9868:
9862:
9859:
9854:
9850:
9847:
9844:
9841:
9838:
9833:
9828:
9824:
9788:
9761:
9758:
9752:
9749:
9742:
9738:
9734:
9731:
9725:
9719:
9716:
9709:
9705:
9701:
9698:
9692:
9686:
9683:
9676:
9672:
9668:
9665:
9659:
9656:
9653:
9649:
9643:
9640:
9635:
9631:
9624:
9620:
9616:
9613:
9576:
9568:
9565:
9562:
9558:
9554:
9551:
9545:
9542:
9537:
9533:
9529:
9526:
9521:
9516:
9512:
9486:
9478:
9473:
9466:
9462:
9459:
9456:
9452:
9448:
9445:
9440:
9435:
9431:
9427:
9424:
9421:
9418:
9414:
9409:
9405:
9402:
9399:
9395:
9391:
9388:
9383:
9378:
9372:
9369:
9364:
9360:
9353:
9346:
9341:
9336:
9332:
9328:
9325:
9292:
9248:
9245:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9220:
9214:
9211:
9206:
9202:
9197:
9194:
9189:
9184:
9180:
9173:
9162:
9132:
9125:
9119:
9115:
9107:
9104:
9097:
9093:
9090:
9087:
9084:
9081:
9078:
9068:
9063:
9059:
9055:
9052:
9049:
9046:
9043:
9040:
9037:
9034:
9031:
8984:
8978:
8975:
8970:
8965:
8961:
8953:
8950:
8944:
8939:
8934:
8930:
8922:
8919:
8913:
8908:
8904:
8900:
8897:
8892:
8888:
8884:
8881:
8878:
8875:
8872:
8869:
8866:
8863:
8842:
8784:
8778:
8775:
8772:
8767:
8764:
8760:
8756:
8753:
8750:
8745:
8741:
8735:
8731:
8724:
8721:
8717:
8712:
8707:
8703:
8697:
8693:
8686:
8683:
8679:
8674:
8671:
8668:
8665:
8660:
8655:
8631:
8630:
8619:
8611:
8603:
8600:
8597:
8594:
8591:
8586:
8581:
8571:
8563:
8560:
8557:
8554:
8548:
8542:
8539:
8536:
8533:
8530:
8527:
8519:
8511:
8508:
8505:
8502:
8499:
8494:
8489:
8479:
8471:
8463:
8455:
8452:
8449:
8446:
8440:
8437:
8434:
8431:
8423:
8418:
8413:
8408:
8372:
8358:
8344:
8329:
8313:
8294:
8285:
8277:
8233:
8222:shift operator
8213:
8199:
8193:
8190:
8187:
8182:
8178:
8174:
8169:
8165:
8149:
8131:
8128:
8116:
8113:
8110:
8107:
8104:
8101:
8098:
8095:
8089:
8084:
8081:
8076:
8068:
8065:
8062:
8058:
8054:
8051:
8048:
8043:
8038:
8035:
8032:
8028:
8021:
8016:
8012:
8008:
8005:
8002:
7995:
7987:
7984:
7981:
7977:
7973:
7970:
7967:
7964:
7961:
7943:Selberg series
7896:
7892:
7889:
7886:
7882:
7879:
7876:
7873:
7870:
7867:
7864:
7861:
7858:
7847:scalar product
7834:
7830:
7824:
7820:
7816:
7813:
7809:
7805:
7800:
7796:
7792:
7787:
7784:
7781:
7777:
7771:
7768:
7765:
7759:
7755:
7731:
7728:
7725:
7721:
7717:
7712:
7708:
7704:
7700:
7696:
7692:
7689:
7685:
7682:
7679:
7675:
7672:
7669:
7665:
7662:
7659:
7656:
7652:
7647:
7638:
7634:
7630:
7625:
7622:
7619:
7615:
7607:
7604:
7601:
7598:
7595:
7591:
7587:
7582:
7579:
7576:
7573:
7570:
7567:
7564:
7560:
7553:
7548:
7545:
7542:
7538:
7533:
7528:
7524:
7520:
7515:
7512:
7509:
7505:
7466:
7463:
7460:
7457:
7454:
7451:
7448:
7445:
7442:
7439:
7436:
7433:
7430:
7427:
7425:
7423:
7420:
7414:
7410:
7407:
7404:
7401:
7398:
7395:
7392:
7389:
7386:
7383:
7376:
7373:
7370:
7364:
7360:
7357:
7354:
7347:
7344:
7341:
7338:
7335:
7332:
7329:
7326:
7324:
7322:
7319:
7316:
7313:
7310:
7305:
7301:
7297:
7289:
7286:
7279:
7275:
7269:
7265:
7261:
7258:
7255:
7248:
7243:
7239:
7235:
7228:
7225:
7218:
7214:
7208:
7204:
7200:
7197:
7194:
7187:
7182:
7178:
7174:
7171:
7168:
7163:
7159:
7155:
7152:
7150:
7148:
7145:
7142:
7139:
7136:
7135:
7131:
7123:
7120:
7118:
7115:
7114:
7110:
7107:
7104:
7099:
7096:
7091:
7088:
7085:
7083:
7080:
7079:
7073:
7070:
7065:
7062:
7055:
7052:
7047:
7045:
7042:
7041:
7036:
7032:
7028:
7024:
7020:
7016:
7012:
7008:
7004:
7001:
6998:
6996:
6993:
6992:
6989:
6988:
6975:
6964:= 2, 2, 4, ...
6923:
6918:
6914:
6910:
6907:
6901:
6898:
6895:
6891:
6887:
6884:
6878:
6873:
6870:
6865:
6857:
6852:
6849:
6846:
6842:
6838:
6833:
6829:
6825:
6822:
6819:
6816:
6781:
6756:
6753:
6750:
6747:
6744:
6741:
6738:
6735:
6732:
6729:
6726:
6723:
6720:
6717:
6714:
6711:
6708:
6705:
6702:
6699:
6694:
6690:
6686:
6683:
6656:
6653:
6646:
6642:
6638:
6635:
6629:
6623:
6618:
6615:
6610:
6587:
6567:
6525:
6522:
6519:
6516:
6513:
6510:
6507:
6502:
6498:
6490:
6485:
6481:
6478:
6475:
6469:
6461:
6456:
6453:
6450:
6446:
6442:
6437:
6433:
6429:
6426:
6423:
6420:
6413:
6410:
6405:
6402:
6399:
6396:
6393:
6390:
6385:
6381:
6372:
6367:
6364:
6361:
6357:
6353:
6350:
6347:
6344:
6341:
6331:
6307:, named after
6294:
6291:
6252:Main article:
6249:
6246:
6245:
6244:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6204:
6201:
6198:
6193:
6189:
6185:
6182:
6179:
6176:
6173:
6168:
6163:
6159:
6152:
6147:
6144:
6139:
6131:
6126:
6123:
6120:
6116:
6112:
6109:
6106:
6103:
6100:
6097:
6094:
6089:
6084:
6080:
6065:
6054:
6050:
6046:
6041:
6037:
6033:
6030:
6027:
6024:
6019:
6015:
6011:
6008:
6003:
5999:
5995:
5992:
5989:
5986:
5982:
5976:
5971:
5967:
5961:
5958:
5955:
5950:
5947:
5942:
5938:
5933:
5929:
5924:
5921:
5918:
5913:
5910:
5905:
5901:
5896:
5890:
5887:
5884:
5879:
5876:
5871:
5867:
5862:
5858:
5855:
5852:
5849:
5846:
5843:
5838:
5833:
5830:
5826:
5805:
5802:
5782:Main article:
5779:
5776:
5764:
5761:
5758:
5755:
5752:
5747:
5743:
5739:
5736:
5731:
5727:
5723:
5706:
5705:
5700:
5696:
5695:
5690:
5686:
5685:
5680:
5676:
5675:
5670:
5666:
5665:
5660:
5656:
5655:
5648:
5608:
5605:
5602:
5599:
5596:
5593:
5590:
5585:
5581:
5577:
5574:
5571:
5568:
5555:
5554:
5551:
5546:
5542:
5541:
5538:
5533:
5529:
5528:
5525:
5520:
5516:
5515:
5512:
5507:
5503:
5502:
5497:
5493:
5492:
5484:
5477:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5410:
5406:
5402:
5399:
5396:
5393:
5390:
5377:
5376:
5373:
5370:
5365:
5361:
5360:
5357:
5354:
5349:
5345:
5344:
5341:
5338:
5333:
5329:
5328:
5325:
5320:
5316:
5315:
5310:
5306:
5305:
5297:
5289:
5284:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5201:
5197:
5193:
5190:
5187:
5184:
5156:
5155:
5152:
5149:
5146:
5143:
5139:
5138:
5135:
5132:
5129:
5126:
5122:
5121:
5118:
5115:
5112:
5108:
5107:
5104:
5101:
5097:
5096:
5093:
5089:
5088:
5080:
5072:
5064:
5057:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4979:
4976:
4973:
4970:
4967:
4932:
4931:
4928:
4924:
4923:
4920:
4916:
4915:
4912:
4908:
4907:
4904:
4900:
4899:
4896:
4892:
4891:
4886:
4851:
4848:
4833:
4830:
4825:
4821:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4791:
4788:
4785:
4781:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4753:
4750:
4747:
4742:
4739:
4736:
4731:
4727:
4649:
4646:
4643:
4640:
4637:
4629:
4624:
4621:
4618:
4614:
4610:
4607:
4604:
4601:
4598:
4590:
4585:
4582:
4579:
4575:
4571:
4568:
4563:
4559:
4555:
4552:
4549:
4546:
4538:
4533:
4530:
4527:
4523:
4519:
4516:
4513:
4510:
4507:
4504:
4499:
4495:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4450:
4446:
4406:
4403:
4400:
4397:
4394:
4391:
4386:
4383:
4380:
4376:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4346:
4343:
4340:
4335:
4331:
4319: − 1
4298: − 1
4279:
4278:Inductive step
4276:
4261:
4258:
4253:
4249:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4120:
4116:
4088:
4085:
4083:
4080:
4055:
4052:
4047:
4043:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4013:
4008:
4004:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3945:
3942:
3939:
3935:
3931:
3928:
3923:
3919:
3915:
3912:
3909:
3906:
3903:
3900:
3843:
3840:
3800:
3797:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3726:
3723:
3718:
3714:
3711:
3708:
3705:
3702:
3699:
3694:
3689:
3685:
3679:
3676:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3648:
3644:
3614:
3610:
3605:
3601:
3597:
3593:
3590:
3583:
3579:
3574:
3571:
3568:
3565:
3562:
3559:
3554:
3549:
3545:
3538:
3535:
3532:
3529:
3526:
3523:
3516:
3512:
3507:
3504:
3501:
3498:
3495:
3492:
3487:
3482:
3478:
3471:
3468:
3465:
3462:
3459:
3456:
3449:
3445:
3440:
3437:
3434:
3431:
3428:
3425:
3420:
3415:
3411:
3404:
3401:
3398:
3395:
3387:
3383:
3379:
3374:
3369:
3365:
3312:
3306:
3297:
3290:
3280:
3277:
3271:
3268:
3265:
3262:
3257:
3253:
3226:
3217:
3210:
3200:
3197:
3191:
3188:
3185:
3182:
3177:
3173:
3136:
3135:
3128:
3121:
3120:
3112:
3111:
3100:
3096:
3092:
3088:
3084:
3081:
3076:
3073:
3067:
3063:
3060:
3056:
3052:
3046:
3041:
3038:
3033:
3025:
3021:
3017:
3014:
3011:
3006:
3001:
2998:
2995:
2991:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2964:
2959:
2955:
2944:
2941:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2903:
2898:
2895:
2890:
2882:
2878:
2874:
2871:
2868:
2863:
2858:
2855:
2852:
2848:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2821:
2816:
2812:
2801:
2798:
2787:
2782:
2777:
2774:
2771:
2768:
2763:
2758:
2752:
2747:
2744:
2739:
2731:
2728:
2725:
2721:
2717:
2714:
2711:
2706:
2701:
2698:
2695:
2691:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2664:
2659:
2655:
2644:
2630:
2629:
2618:
2611:
2607:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2539:
2534:
2528:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2479:
2474:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2431:
2424:
2420:
2415:
2412:
2409:
2406:
2403:
2400:
2395:
2390:
2386:
2379:
2376:
2373:
2370:
2366:
2363:
2352:
2349:
2338:
2331:
2327:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2259:
2254:
2248:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2208:
2203:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2151:
2144:
2140:
2135:
2132:
2129:
2126:
2123:
2120:
2115:
2110:
2106:
2099:
2096:
2093:
2090:
2086:
2083:
2072:
2065:
2064:
2053:
2046:
2042:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1977:
1972:
1966:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1926:
1921:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1878:
1871:
1867:
1862:
1859:
1856:
1853:
1850:
1847:
1842:
1837:
1833:
1826:
1823:
1820:
1817:
1813:
1810:
1799:
1717:
1716:
1631:
1629:
1622:
1616:
1613:
1541:
1537:
1532:
1528:
1524:
1520:
1517:
1514:
1511:
1508:
1504:
1501:
1497:
1492:
1488:
1485:
1482:
1479:
1476:
1473:
1468:
1464:
1434:
1431:
1428:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1388:
1385:
1381:
1376:
1372:
1369:
1366:
1363:
1360:
1357:
1352:
1348:
1322:
1319:
1316:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1276:
1273:
1269:
1264:
1260:
1257:
1254:
1251:
1248:
1245:
1240:
1236:
1194:
1189:
1185:
1182:
1179:
1176:
1173:
1170:
1165:
1161:
1154:
1149:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1086:
1081:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1039:
1036:
1033:
1029:
1025:
1022:
1019:
1016:
1012:
1009:
988:of a function
974:
971:
959:
956:
953:
950:
947:
944:
941:
936:
932:
928:
924:
920:
917:
914:
911:
908:
905:
902:
897:
893:
889:
885:
881:
878:
872:
869:
863:
860:
857:
854:
851:
848:
842:
839:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
798:
794:
763:
760:
757:
754:
751:
748:
745:
742:
739:
734:
730:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
667:
663:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
545:
541:
537:
534:
531:
528:
525:
522:
519:
516:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
427:
423:
395:
392:
389:
386:
381:
377:
325:
322:
318:infinitesimals
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
152:
149:
146:
143:
115:
28:
9:
6:
4:
3:
2:
14097:
14086:
14083:
14081:
14078:
14076:
14073:
14071:
14068:
14066:
14063:
14061:
14058:
14056:
14053:
14052:
14050:
14035:
14032:
14030:
14027:
14025:
14022:
14020:
14017:
14015:
14012:
14010:
14007:
14003:
14000:
13998:
13995:
13993:
13990:
13988:
13985:
13983:
13980:
13979:
13977:
13973:
13970:
13969:
13967:
13966:
13964:
13960:
13950:
13947:
13945:
13942:
13941:
13938:
13930:
13927:
13925:
13922:
13920:
13917:
13916:
13915:
13912:
13908:
13905:
13904:
13903:
13900:
13898:
13895:
13893:
13890:
13888:
13885:
13883:
13880:
13879:
13877:
13875:
13871:
13868:
13864:
13858:
13857:
13853:
13851:
13850:
13846:
13844:
13841:
13839:
13836:
13834:
13831:
13829:
13826:
13824:
13821:
13819:
13818:Infinitesimal
13816:
13814:
13811:
13809:
13806:
13804:
13801:
13799:
13796:
13794:
13791:
13790:
13788:
13786:
13782:
13776:
13773:
13771:
13768:
13766:
13763:
13761:
13758:
13756:
13753:
13752:
13750:
13744:
13736:
13733:
13731:
13728:
13726:
13723:
13721:
13718:
13716:
13713:
13711:
13708:
13706:
13703:
13701:
13698:
13696:
13693:
13691:
13688:
13687:
13685:
13681:
13678:
13674:
13671:
13669:
13666:
13665:
13664:
13661:
13659:
13656:
13654:
13651:
13649:
13646:
13644:
13641:
13639:
13636:
13634:
13631:
13630:
13628:
13626:
13623:
13622:
13620:
13616:
13608:
13605:
13603:
13600:
13598:
13595:
13593:
13590:
13589:
13587:
13585:
13582:
13580:
13577:
13575:
13572:
13570:
13567:
13565:
13562:
13560:
13559:Line integral
13557:
13555:
13552:
13550:
13547:
13545:
13542:
13540:
13537:
13535:
13532:
13531:
13529:
13527:
13523:
13515:
13512:
13510:
13507:
13505:
13502:
13500:
13497:
13496:
13494:
13490:
13487:
13485:
13482:
13480:
13477:
13475:
13472:
13470:
13467:
13466:
13464:
13463:
13461:
13459:
13455:
13449:
13446:
13444:
13441:
13437:
13434:
13432:
13431:Washer method
13429:
13428:
13426:
13424:
13421:
13417:
13414:
13413:
13412:
13409:
13405:
13402:
13400:
13397:
13395:
13394:trigonometric
13392:
13391:
13390:
13387:
13385:
13382:
13378:
13375:
13374:
13373:
13370:
13368:
13365:
13363:
13360:
13358:
13355:
13353:
13350:
13348:
13345:
13344:
13342:
13340:
13336:
13328:
13325:
13323:
13320:
13318:
13315:
13314:
13313:
13310:
13306:
13303:
13301:
13298:
13297:
13295:
13291:
13288:
13286:
13283:
13281:
13278:
13276:
13273:
13272:
13271:
13268:
13264:
13263:Related rates
13261:
13259:
13256:
13254:
13251:
13249:
13246:
13245:
13243:
13239:
13236:
13232:
13229:
13228:
13227:
13224:
13222:
13219:
13217:
13214:
13212:
13209:
13207:
13204:
13202:
13199:
13198:
13197:
13194:
13190:
13187:
13185:
13182:
13181:
13180:
13177:
13175:
13172:
13170:
13167:
13165:
13162:
13160:
13157:
13155:
13152:
13150:
13147:
13146:
13144:
13142:
13138:
13132:
13129:
13127:
13124:
13122:
13119:
13115:
13112:
13111:
13110:
13107:
13105:
13102:
13101:
13099:
13097:
13093:
13087:
13084:
13082:
13079:
13077:
13074:
13072:
13069:
13067:
13064:
13062:
13059:
13057:
13054:
13052:
13049:
13047:
13044:
13042:
13039:
13037:
13034:
13032:
13029:
13027:
13024:
13023:
13021:
13019:
13015:
13011:
13004:
12999:
12997:
12992:
12990:
12985:
12984:
12981:
12975:
12972:
12970:
12969:
12964:
12962:
12959:
12955:
12951:
12950:
12945:
12941:
12940:
12931:
12927:
12923:
12919:
12917:
12913:
12909:
12908:
12900:
12896:
12892:
12888:
12881:
12877:
12870:
12862:
12858:
12852:
12844:
12842:0-12-056760-1
12838:
12834:
12827:
12819:
12817:0-486-67260-3
12813:
12809:
12802:
12794:
12790:
12785:
12780:
12776:
12772:
12767:
12762:
12758:
12754:
12750:
12743:
12735:
12731:
12727:
12723:
12719:
12715:
12710:
12705:
12701:
12697:
12693:
12687:
12679:
12673:
12669:
12668:
12660:
12651:
12647:
12646:
12641:
12640:Boole, George
12635:
12628:
12624:
12619:
12611:
12607:
12602:
12597:
12593:
12589:
12584:
12579:
12575:
12571:
12567:
12560:
12551:
12546:
12542:
12538:
12531:
12524:
12518:
12516:
12509:
12503:
12499:
12494:
12488:
12483:
12477:
12472:
12457:
12453:
12449:
12445:
12441:
12434:
12426:
12420:
12414:
12410:
12406:
12400:
12392:
12386:
12382:
12375:
12373:
12371:
12362:
12356:
12352:
12348:
12342:
12340:
12338:
12329:
12323:
12319:
12314:
12313:
12304:
12302:
12300:
12295:
12285:
12282:
12280:
12277:
12275:
12272:
12270:
12267:
12265:
12262:
12260:
12257:
12255:
12252:
12249:
12246:
12244:
12241:
12239:
12236:
12235:
12228:
12224:
12220:
12216:
12212:
12208:
12201:
12197:
12193:
12189:
12185:
12168:
12162:
12159:
12156:
12148:
12145:
12142:
12139:
12136:
12133:
12130:
12124:
12121:
12115:
12112:
12109:
12106:
12103:
12097:
12094:
12088:
12085:
12082:
12079:
12076:
12070:
12067:
12061:
12058:
12055:
12049:
12046:
12043:
12037:
12034:
12031:
12028:
12025:
12019:
12016:
12010:
12007:
12004:
12001:
11998:
11992:
11989:
11983:
11980:
11977:
11974:
11971:
11968:
11965:
11959:
11953:
11947:
11944:
11941:
11933:
11930:
11926:
11912:
11895:
11889:
11886:
11883:
11875:
11872:
11869:
11866:
11863:
11860:
11857:
11851:
11848:
11842:
11839:
11836:
11833:
11830:
11827:
11824:
11818:
11815:
11809:
11806:
11803:
11800:
11797:
11794:
11791:
11785:
11782:
11776:
11773:
11770:
11767:
11764:
11761:
11758:
11752:
11746:
11744:
11736:
11733:
11730:
11722:
11719:
11715:
11703:
11699:
11691:
11688:
11685:
11682:
11679:
11673:
11670:
11664:
11661:
11658:
11652:
11649:
11646:
11640:
11637:
11634:
11631:
11628:
11622:
11616:
11614:
11606:
11603:
11600:
11592:
11589:
11585:
11573:
11569:
11561:
11558:
11555:
11552:
11549:
11543:
11540:
11534:
11531:
11528:
11522:
11519:
11516:
11510:
11507:
11504:
11501:
11498:
11492:
11486:
11484:
11476:
11473:
11470:
11462:
11459:
11455:
11444:
11441:
11433:
11430:
11427:
11424:
11421:
11415:
11412:
11406:
11403:
11400:
11397:
11394:
11388:
11382:
11380:
11372:
11369:
11366:
11358:
11354:
11343:
11340:
11332:
11329:
11326:
11323:
11320:
11314:
11311:
11305:
11302:
11299:
11296:
11293:
11287:
11281:
11279:
11271:
11268:
11265:
11257:
11253:
11239:
11237:
11213:
11209:
11205:
11201:
11198:
11194:
11190:
11186:
11181:
11178:
11172:
11168:
11164:
11151:
11145:
11141:
11134:
11121:
11113:
11109:
11103:
11093:
11089:
11072:
11066:
11063:
11060:
11057:
11051:
11046:
11042:
11036:
11031:
11028:
11025:
11021:
11017:
11011:
11002:
10994:
10989:
10976:
10972:
10971:
10965:
10938:
10935:
10932:
10926:
10923:
10917:
10911:
10908:
10906:
10898:
10892:
10884:
10879:
10876:
10873:
10869:
10858:
10852:
10849:
10843:
10840:
10837:
10831:
10828:
10826:
10818:
10812:
10804:
10799:
10796:
10793:
10789:
10773:
10772:
10768:
10745:
10739:
10736:
10730:
10727:
10722:
10715:
10712:
10709:
10702:
10696:
10692:
10687:
10684:
10679:
10654:
10648:
10642:
10637:
10630:
10622:
10616:
10608:
10605:
10601:
10592:
10586:
10580:
10575:
10568:
10560:
10551:
10542:
10535:
10531:
10526:
10523:
10518:
10503:
10502:
10501:Quotient rule
10498:
10477:
10470:
10464:
10461:
10454:
10451:
10448:
10441:
10438:
10436:
10428:
10425:
10412:
10405:
10399:
10396:
10389:
10386:
10383:
10376:
10373:
10371:
10363:
10360:
10339:
10338:
10334:
10333:
10332:
10297:
10288:
10285:
10282:
10273:
10270:
10264:
10258:
10255:
10252:
10246:
10229:
10217:
10216:
10212:
10195:
10192:
10189:
10175:
10167:
10166:Constant rule
10164:
10163:
10162:
10160:
10157:Analogous to
10150:
10148:
10143:
10141:
10137:
10121:
10109:
10106:
10103:
10097:
10091:
10086:
10080:
10077:
10072:
10069:
10065:
10059:
10056:
10050:
10046:
10043:
10031:
10025:
10015:
10011:
10006:
10004:
10000:
9996:
9977:
9967:
9964:
9956:
9953:
9950:
9947:
9941:
9938:
9934:
9930:
9927:
9920:
9917:
9909:
9906:
9903:
9900:
9894:
9891:
9887:
9881:
9876:
9866:
9860:
9857:
9848:
9845:
9842:
9839:
9831:
9826:
9809:
9808:
9807:
9800:
9791:
9780:
9775:
9759:
9756:
9750:
9747:
9740:
9732:
9723:
9717:
9714:
9707:
9699:
9690:
9684:
9681:
9674:
9666:
9657:
9654:
9651:
9647:
9641:
9638:
9633:
9622:
9618:
9614:
9611:
9599:
9595:
9591:
9574:
9566:
9563:
9560:
9552:
9543:
9540:
9535:
9527:
9519:
9514:
9484:
9471:
9464:
9460:
9457:
9454:
9450:
9446:
9443:
9433:
9429:
9425:
9422:
9419:
9416:
9412:
9407:
9403:
9400:
9397:
9393:
9389:
9386:
9381:
9376:
9370:
9367:
9362:
9351:
9344:
9339:
9334:
9326:
9312:
9303:
9295:
9287:
9283:
9276:
9273: thus
9270:
9266:
9259:
9246:
9243:
9240:
9234:
9231:
9222:
9218:
9212:
9209:
9204:
9195:
9192:
9187:
9182:
9171:
9161:
9157:
9152:
9148:
9143:
9130:
9123:
9117:
9113:
9105:
9102:
9095:
9091:
9088:
9085:
9082:
9076:
9061:
9053:
9050:
9044:
9041:
9038:
9035:
9029:
9020:
9018:
9017:
9010:
9006:
9001:
8996:
8982:
8976:
8973:
8968:
8963:
8951:
8948:
8942:
8937:
8932:
8920:
8917:
8911:
8906:
8898:
8890:
8882:
8879:
8873:
8870:
8867:
8861:
8850:
8845:
8829:
8825:
8817:
8809:
8782:
8776:
8770:
8762:
8758:
8754:
8751:
8748:
8743:
8733:
8729:
8722:
8719:
8715:
8710:
8705:
8695:
8691:
8684:
8681:
8677:
8672:
8666:
8663:
8658:
8639:
8638:Taylor series
8634:
8617:
8598:
8592:
8589:
8584:
8558:
8552:
8546:
8537:
8534:
8531:
8525:
8506:
8500:
8497:
8492:
8469:
8450:
8444:
8435:
8429:
8416:
8411:
8394:
8393:
8392:
8390:
8385:
8380:
8375:
8370:
8366:
8361:
8356:
8352:
8347:
8338:
8332:
8324:
8316:
8297:
8288:
8280:
8267:
8265:
8253:
8249:
8245:
8241:
8236:
8228:, defined by
8223:
8216:
8197:
8191:
8185:
8180:
8172:
8167:
8152:
8141:
8138:, called the
8137:
8127:
8114:
8108:
8105:
8102:
8099:
8093:
8082:
8079:
8066:
8063:
8060:
8052:
8049:
8041:
8036:
8033:
8030:
8026:
8014:
8010:
8006:
8003:
8000:
7985:
7982:
7979:
7975:
7971:
7965:
7959:
7950:
7948:
7944:
7940:
7935:
7933:
7929:
7919:
7915:
7914:-adic numbers
7907:
7894:
7890:
7887:
7884:
7880:
7877:
7874:
7868:
7862:
7856:
7848:
7832:
7828:
7822:
7818:
7814:
7811:
7807:
7803:
7798:
7794:
7790:
7785:
7782:
7779:
7775:
7769:
7766:
7763:
7757:
7753:
7729:
7726:
7723:
7715:
7710:
7706:
7698:
7694:
7690:
7687:
7683:
7680:
7677:
7670:
7667:
7663:
7660:
7657:
7654:
7650:
7645:
7636:
7632:
7628:
7623:
7620:
7617:
7613:
7605:
7602:
7599:
7596:
7593:
7585:
7580:
7577:
7574:
7571:
7568:
7565:
7562:
7551:
7546:
7543:
7540:
7531:
7526:
7522:
7518:
7513:
7510:
7507:
7494:
7485:
7461:
7458:
7455:
7446:
7443:
7440:
7434:
7431:
7428:
7426:
7412:
7405:
7402:
7399:
7390:
7387:
7384:
7374:
7371:
7368:
7362:
7358:
7355:
7352:
7345:
7342:
7339:
7336:
7333:
7330:
7327:
7325:
7311:
7308:
7303:
7299:
7287:
7284:
7277:
7267:
7263:
7259:
7256:
7246:
7241:
7233:
7226:
7223:
7216:
7206:
7202:
7198:
7195:
7185:
7180:
7172:
7169:
7166:
7161:
7153:
7151:
7143:
7137:
7121:
7116:
7108:
7097:
7094:
7086:
7081:
7071:
7068:
7053:
7050:
7043:
7034:
7022:
7010:
7002:
6999:
6994:
6974:
6969:
6963:
6959:
6954:
6952:
6948:
6943:
6941:
6937:
6921:
6916:
6908:
6899:
6896:
6893:
6885:
6871:
6868:
6855:
6850:
6847:
6844:
6840:
6836:
6831:
6823:
6820:
6817:
6806:
6802:
6797:
6793:
6789:
6779:
6774:
6773:empty product
6770:
6751:
6748:
6745:
6742:
6739:
6733:
6727:
6724:
6721:
6712:
6709:
6706:
6700:
6697:
6692:
6684:
6673:
6654:
6651:
6644:
6636:
6627:
6616:
6613:
6585:
6565:
6558:
6550:
6542:
6536:
6523:
6517:
6508:
6500:
6483:
6479:
6476:
6473:
6454:
6451:
6448:
6444:
6440:
6435:
6427:
6424:
6421:
6411:
6408:
6400:
6391:
6383:
6365:
6362:
6359:
6355:
6351:
6345:
6339:
6330:
6328:
6327:
6322:
6321:James Gregory
6318:
6315:(named after
6314:
6310:
6306:
6302:
6301:
6300:Newton series
6290:
6288:
6284:
6279:
6277:
6273:
6269:
6265:
6261:
6255:
6231:
6225:
6222:
6219:
6216:
6213:
6210:
6202:
6199:
6196:
6191:
6180:
6177:
6174:
6166:
6161:
6145:
6142:
6129:
6124:
6121:
6118:
6110:
6104:
6101:
6098:
6095:
6087:
6082:
6069:
6066:
6052:
6048:
6044:
6039:
6035:
6031:
6028:
6025:
6022:
6017:
6013:
6009:
6006:
6001:
5997:
5993:
5990:
5987:
5984:
5980:
5974:
5969:
5959:
5956:
5953:
5948:
5945:
5940:
5936:
5927:
5922:
5919:
5916:
5911:
5908:
5903:
5899:
5888:
5885:
5882:
5877:
5874:
5869:
5865:
5856:
5850:
5847:
5844:
5836:
5831:
5828:
5808:
5807:
5801:
5798:
5791:
5785:
5775:
5762:
5759:
5756:
5753:
5750:
5745:
5741:
5737:
5734:
5729:
5725:
5721:
5701:
5698:
5697:
5691:
5688:
5687:
5681:
5678:
5677:
5671:
5668:
5667:
5661:
5658:
5657:
5653:
5649:
5646:
5642:
5641:
5638:
5635:
5634:
5625:
5619:
5606:
5603:
5600:
5597:
5594:
5591:
5588:
5583:
5579:
5575:
5572:
5569:
5566:
5552:
5547:
5544:
5543:
5539:
5534:
5531:
5530:
5526:
5521:
5518:
5517:
5513:
5508:
5505:
5504:
5498:
5495:
5494:
5490:
5485:
5482:
5478:
5475:
5471:
5470:
5467:
5464:
5461:
5450:
5444:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5408:
5404:
5400:
5397:
5394:
5391:
5388:
5374:
5371:
5366:
5363:
5362:
5358:
5355:
5350:
5347:
5346:
5342:
5339:
5334:
5331:
5330:
5326:
5321:
5318:
5317:
5311:
5308:
5307:
5303:
5298:
5295:
5290:
5285:
5280:
5279:
5276:
5273:
5270:
5259:
5253:
5240:
5237:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5199:
5195:
5191:
5188:
5185:
5182:
5168:
5153:
5150:
5147:
5144:
5141:
5140:
5136:
5133:
5130:
5127:
5124:
5123:
5119:
5116:
5113:
5110:
5109:
5105:
5102:
5099:
5098:
5094:
5091:
5090:
5086:
5081:
5078:
5073:
5070:
5065:
5062:
5058:
5055:
5051:
5050:
5047:
5044:
5028:
5025:
5022:
5016:
5010:
5007:
5004:
5001:
4998:
4992:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4951:
4947:
4940:
4929:
4926:
4925:
4921:
4918:
4917:
4913:
4910:
4909:
4905:
4902:
4901:
4897:
4894:
4893:
4890:
4887:
4885:
4882:
4881:
4878:
4874:
4869:
4863:
4859:
4847:
4844:
4831:
4828:
4823:
4819:
4815:
4812:
4809:
4803:
4800:
4797:
4789:
4786:
4783:
4779:
4775:
4772:
4769:
4766:
4763:
4757:
4748:
4740:
4737:
4734:
4729:
4714:
4710:
4704:
4697:
4691:
4684:
4677:
4673:
4666:
4660:
4644:
4638:
4635:
4627:
4622:
4619:
4616:
4612:
4608:
4605:
4602:
4599:
4588:
4583:
4580:
4577:
4573:
4569:
4566:
4561:
4557:
4553:
4547:
4536:
4531:
4528:
4525:
4517:
4514:
4511:
4505:
4502:
4497:
4489:
4486:
4483:
4477:
4471:
4465:
4456:
4448:
4434:
4427:
4423:
4417:
4404:
4398:
4395:
4392:
4384:
4381:
4378:
4374:
4370:
4367:
4361:
4352:
4344:
4341:
4338:
4333:
4318:
4311:
4304:
4297:
4290:
4286:
4275:
4272:
4259:
4256:
4251:
4247:
4243:
4240:
4237:
4234:
4231:
4225:
4222:
4219:
4216:
4210:
4204:
4201:
4195:
4192:
4189:
4183:
4177:
4171:
4165:
4162:
4156:
4153:
4150:
4144:
4141:
4135:
4126:
4118:
4099:
4095:
4079:
4072:
4066:
4053:
4050:
4045:
4041:
4037:
4034:
4028:
4019:
4011:
4006:
3991:
3985:
3979:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3943:
3940:
3937:
3933:
3929:
3926:
3921:
3917:
3913:
3910:
3904:
3898:
3889:
3884:
3879:
3872:
3865:
3861:
3854:
3849:
3839:
3836:
3834:
3833:Taylor series
3829:
3822:
3818:
3814:approximates
3798:
3795:
3787:
3781:
3778:
3775:
3769:
3766:
3763:
3757:
3754:
3751:
3745:
3742:
3739:
3736:
3730:
3724:
3721:
3716:
3709:
3700:
3692:
3687:
3677:
3674:
3669:
3663:
3654:
3646:
3625:
3612:
3608:
3603:
3599:
3595:
3591:
3588:
3581:
3577:
3569:
3560:
3552:
3547:
3543:
3536:
3530:
3524:
3521:
3514:
3510:
3502:
3493:
3485:
3480:
3469:
3463:
3457:
3454:
3447:
3443:
3435:
3426:
3418:
3413:
3402:
3396:
3385:
3381:
3377:
3372:
3367:
3363:
3350:
3344:
3340:
3336:
3332:
3328:
3323:
3310:
3295:
3288:
3278:
3275:
3263:
3255:
3251:
3215:
3208:
3198:
3195:
3183:
3175:
3171:
3150:
3144:
3131:
3117:
3098:
3094:
3090:
3086:
3082:
3079:
3074:
3071:
3065:
3061:
3058:
3054:
3050:
3039:
3036:
3023:
3015:
3012:
3004:
2999:
2996:
2993:
2989:
2985:
2979:
2970:
2962:
2957:
2953:
2945:
2942:
2928:
2922:
2919:
2916:
2913:
2907:
2896:
2893:
2880:
2872:
2869:
2861:
2856:
2853:
2850:
2846:
2842:
2836:
2827:
2819:
2814:
2802:
2799:
2785:
2775:
2772:
2769:
2766:
2756:
2745:
2742:
2729:
2726:
2723:
2715:
2712:
2704:
2699:
2696:
2693:
2689:
2685:
2679:
2670:
2662:
2657:
2645:
2642:
2641:
2640:
2638:
2616:
2609:
2605:
2597:
2594:
2591:
2588:
2582:
2579:
2573:
2570:
2567:
2561:
2558:
2555:
2549:
2543:
2537:
2532:
2526:
2519:
2516:
2513:
2510:
2504:
2501:
2495:
2492:
2489:
2483:
2477:
2472:
2465:
2462:
2459:
2453:
2450:
2444:
2438:
2429:
2422:
2418:
2410:
2401:
2393:
2388:
2377:
2371:
2364:
2361:
2353:
2350:
2336:
2329:
2325:
2317:
2311:
2308:
2302:
2299:
2296:
2290:
2287:
2284:
2278:
2275:
2272:
2269:
2263:
2257:
2252:
2246:
2239:
2233:
2230:
2224:
2221:
2218:
2212:
2206:
2201:
2194:
2191:
2188:
2182:
2179:
2173:
2170:
2167:
2164:
2158:
2149:
2142:
2138:
2130:
2121:
2113:
2108:
2097:
2091:
2084:
2081:
2073:
2070:
2069:
2068:
2051:
2044:
2040:
2032:
2029:
2026:
2020:
2017:
2011:
2005:
2002:
1999:
1993:
1990:
1987:
1981:
1975:
1970:
1964:
1957:
1954:
1951:
1945:
1942:
1936:
1930:
1924:
1919:
1912:
1906:
1903:
1897:
1894:
1891:
1885:
1876:
1869:
1865:
1857:
1848:
1840:
1835:
1831:
1824:
1818:
1811:
1808:
1800:
1797:
1796:
1795:
1783:
1766:
1758:
1754:
1737:
1729:
1725:
1713:
1710:
1702:
1691:
1688:
1684:
1681:
1677:
1674:
1670:
1667:
1663:
1660: –
1659:
1655:
1654:Find sources:
1648:
1644:
1638:
1637:
1632:This section
1630:
1626:
1621:
1620:
1612:
1608:
1606:
1598:
1592:
1588:
1577:
1573:
1562:
1558:
1552:
1539:
1535:
1530:
1526:
1522:
1518:
1515:
1509:
1502:
1499:
1495:
1490:
1483:
1474:
1466:
1462:
1445:
1432:
1426:
1417:
1408:
1402:
1399:
1393:
1386:
1383:
1379:
1374:
1367:
1358:
1350:
1333:
1320:
1314:
1305:
1296:
1290:
1287:
1281:
1274:
1271:
1267:
1262:
1255:
1246:
1238:
1218:
1205:
1192:
1187:
1180:
1171:
1163:
1152:
1147:
1140:
1134:
1131:
1125:
1122:
1119:
1113:
1097:
1084:
1079:
1072:
1066:
1063:
1057:
1054:
1051:
1045:
1037:
1031:
1023:
1017:
1010:
1007:
999:
987:
982:
980:
970:
957:
951:
942:
934:
930:
926:
918:
912:
903:
895:
891:
887:
879:
870:
867:
861:
858:
852:
849:
840:
837:
831:
828:
822:
819:
813:
804:
796:
792:
783:
776:Finally, the
774:
761:
755:
752:
749:
740:
732:
724:
718:
715:
712:
706:
703:
697:
691:
688:
682:
673:
665:
647:
643:
637:
633:
624:
615:
602:
596:
590:
587:
581:
578:
575:
569:
566:
560:
551:
543:
535:
529:
520:
497:
484:
478:
472:
469:
463:
460:
457:
451:
448:
442:
433:
425:
409:
393:
387:
379:
366:
357:
355:
351:
347:
339:
335:
330:
321:
319:
315:
311:
300:
294:
289:
288:Károly Jordan
285:
281:
277:
272:
270:
266:
261:
259:
255:
251:
247:
243:
227:
221:
215:
212:
206:
203:
200:
194:
191:
185:
176:
147:
129:
105:
100:
98:
95:, especially
94:
90:
86:
82:
78:
77:
73:, one gets a
71:
67:
60:
56:
52:
48:
44:
40:
35:
27:
19:
13929:Secant cubed
13854:
13847:
13828:Isaac Newton
13798:Brook Taylor
13465:Derivatives
13436:Shell method
13164:Differential
13045:
12967:
12947:
12921:
12911:
12890:
12886:
12869:
12860:
12851:
12832:
12826:
12807:
12801:
12756:
12752:
12742:
12699:
12695:
12686:
12666:
12659:
12644:
12634:
12626:
12618:
12573:
12569:
12559:
12540:
12536:
12523:
12514:
12508:
12493:
12482:
12471:
12459:. Retrieved
12447:
12443:
12433:
12419:
12404:
12399:
12380:
12350:
12311:
12222:
12218:
12214:
12210:
12206:
12199:
12195:
12191:
12187:
12183:
11913:
11240:
11233:
11184:
11170:
11166:
11162:
11149:
11143:
11136:
11129:
11116:
11107:
11098:
11091:
11087:
10974:
10963:
10769:
10499:
10337:Product rule
10335:
10320:, including
10315:
10213:
10165:
10156:
10144:
10007:
9994:
9992:
9798:
9789:
9776:
9600:
9593:
9589:
9304:
9293:
9285:
9281:
9274:
9268:
9264:
9261:
9163:
9155:
9144:
9021:
9014:
9008:
9004:
8997:
8848:
8843:
8827:
8807:
8635:
8632:
8389:Leibniz rule
8386:
8378:
8373:
8368:
8364:
8359:
8354:
8350:
8345:
8330:
8325:
8314:
8295:
8286:
8278:
8268:
8251:
8247:
8243:
8239:
8234:
8214:
8150:
8133:
7951:
7936:
7930:
7908:
7486:
6972:
6961:
6955:
6944:
6798:
6791:
6787:
6777:
6538:
6332:
6324:
6317:Isaac Newton
6312:
6309:Isaac Newton
6304:
6298:
6296:
6280:
6257:
6068:Leibniz rule
5793:
5709:
5651:
5644:
5632:
5630:
5623:
5620:
5558:
5488:
5480:
5473:
5465:
5459:
5448:
5446:Solving for
5445:
5380:
5301:
5293:
5274:
5268:
5257:
5255:Solving for
5254:
5166:
5159:
5084:
5076:
5068:
5060:
5053:
5045:
4949:
4945:
4938:
4935:
4888:
4883:
4872:
4867:
4861:
4857:
4853:
4845:
4712:
4708:
4702:
4695:
4689:
4682:
4675:
4671:
4664:
4661:
4432:
4425:
4421:
4418:
4316:
4309:
4302:
4295:
4288:
4284:
4281:
4273:
4097:
4093:
4090:
4070:
4067:
3989:
3983:
3980:
3887:
3882:
3877:
3870:
3863:
3859:
3852:
3846:For a given
3845:
3837:
3827:
3820:
3816:
3626:
3351:
3343:saddle-point
3324:
3151:
3129:
3113:
2633:
2631:
2066:
1781:
1764:
1756:
1752:
1735:
1727:
1723:
1720:
1705:
1696:
1686:
1679:
1672:
1665:
1653:
1641:Please help
1636:verification
1633:
1609:
1590:
1586:
1575:
1571:
1560:
1556:
1553:
1446:
1334:
1206:
1098:
983:
976:
784:is given by
777:
775:
645:
641:
635:
631:
618:
616:
498:
360:
358:
353:
349:
345:
343:
337:
333:
310:Isaac Newton
286:(1933), and
280:George Boole
276:Brook Taylor
273:
262:
101:
91:solution of
74:
69:
65:
58:
54:
50:
46:
42:
38:
33:
31:
26:
13997:of surfaces
13748:and numbers
13710:Dirichlet's
13680:Telescoping
13633:Alternating
13221:L'Hôpital's
13018:Precalculus
12916:online copy
12347:Peter Olver
12259:FTCS scheme
11208:convolution
10161:, we have:
9151:commutators
5452:, which is
4850:Application
3842:Polynomials
1584:even, then
992:at a point
324:Basic types
306: 1592
291: [
254:derivatives
163:defined by
81:derivatives
14049:Categories
13793:Adequality
13479:Divergence
13352:Arc length
13149:Derivative
12692:Zachos, C.
12583:2008.11139
12576:(1): 007.
12550:1606.02044
12291:References
9993:and hence
9777:As in the
9277:involving
8824:polynomial
8224:with step
6968:polynomial
6541:polynomial
5804:Properties
4680:of degree
3850:of degree
3848:polynomial
1669:newspapers
986:derivative
367:, denoted
299:Jost Bürgi
13992:of curves
13987:Curvature
13874:Integrals
13668:Maclaurin
13648:Geometric
13539:Geometric
13489:Laplacian
13201:linearity
13041:Factorial
12954:EMS Press
12810:. Dover.
12766:1304.0429
12709:0710.2306
12610:221293056
12515:Principia
12461:April 17,
12146:−
12134:−
12113:−
12095:−
12080:−
12068:−
12017:−
11990:−
11954:≈
11873:−
11861:−
11828:−
11816:−
11807:−
11783:−
11747:≈
11689:−
11647:−
11617:≈
11553:−
11517:−
11487:≈
11431:−
11413:−
11383:≈
11324:−
11312:−
11282:≈
11043:μ
11022:∑
10995:μ
10986:Δ
10936:−
10924:−
10890:∇
10870:∑
10850:−
10810:Δ
10790:∑
10743:∇
10740:−
10731:⋅
10720:∇
10713:−
10707:∇
10676:∇
10628:∇
10609:⋅
10566:∇
10558:∇
10515:∇
10475:∇
10468:∇
10465:−
10459:∇
10446:∇
10420:∇
10410:Δ
10403:Δ
10394:Δ
10381:Δ
10355:Δ
10295:Δ
10280:Δ
10241:Δ
10228:constants
10215:Linearity
10187:Δ
10098:π
10057:π
10047:
10038:↦
10026:δ
9954:λ
9942:
9931:λ
9907:λ
9895:
9873:Δ
9846:λ
9823:Δ
9760:⋯
9724:−
9658:−
9639:−
9615:
9564:−
9511:Δ
9497:so that
9458:−
9447:−
9434:⋯
9420:−
9401:−
9368:−
9340:≡
9284: (
9210:−
9179:Δ
9160:limits),
9114:δ
9092:
9058:∇
9054:−
9045:
9039:−
8977:⋯
8974:−
8960:Δ
8929:Δ
8912:−
8903:Δ
8887:Δ
8874:
8777:
8771:−
8752:⋯
8654:Δ
8590:
8580:Δ
8498:
8488:Δ
8417:
8407:Δ
8192:
8186:−
8164:Δ
8064:−
8050:−
8027:∑
8004:−
7976:∑
7888:ξ
7878:⋅
7872:Δ
7863:ξ
7815:−
7812:ξ
7804:⋅
7720:Δ
7703:Δ
7688:−
7671:≤
7646:∋
7629:−
7603:−
7590:Δ
7586:−
7578:−
7559:Δ
7537:Δ
7504:Δ
7459:−
7444:−
7403:−
7388:−
7375:⋅
7356:−
7346:⋅
7334:⋅
7260:−
7247:⋅
7238:Δ
7199:−
7186:⋅
7177:Δ
7167:⋅
7158:Δ
7098:_
7072:_
7054:_
7031:Δ
7019:Δ
7007:Δ
6897:−
6841:∑
6743:−
6734:⋯
6725:−
6710:−
6586:π
6566:π
6543:function
6497:Δ
6477:−
6460:∞
6445:∑
6425:−
6380:Δ
6371:∞
6356:∑
6200:−
6188:Δ
6158:Δ
6115:∑
6079:Δ
6029:⋯
5966:Δ
5957:−
5932:∑
5928:⋯
5920:−
5895:∑
5886:−
5861:∑
5825:Δ
5760:−
5735:−
5604:⋅
5589:⋅
5576:⋅
5429:⋅
5414:⋅
5401:⋅
5389:−
5238:⋅
5226:⋅
5220:⋅
5205:⋅
5192:⋅
5017:−
4875:≠ 0
4801:−
4787:−
4776:⋅
4738:−
4726:Δ
4667:≠ 0
4620:−
4581:−
4548:−
4529:−
4445:Δ
4396:−
4382:−
4342:−
4330:Δ
4312:≠ 0
4305:≥ 2
4211:−
4163:−
4115:Δ
4087:Base case
4003:Δ
3992:≠ 0
3941:−
3873:≠ 0
3855:≥ 1
3819: ′(
3752:−
3725:−
3684:Δ
3670:−
3643:Δ
3544:δ
3477:∇
3410:Δ
3252:δ
3199:−
3172:δ
3080:−
3013:−
2990:∑
2954:δ
2917:−
2870:−
2847:∑
2811:∇
2727:−
2713:−
2690:∑
2654:Δ
2592:−
2571:−
2556:−
2514:−
2502:−
2493:−
2478:−
2463:−
2451:−
2385:∇
2378:≈
2285:−
2231:−
2207:−
2180:−
2105:Δ
2098:≈
2030:−
2000:−
1955:−
1943:−
1925:−
1904:−
1832:δ
1825:≈
1755: ′(
1726: ′(
1699:July 2018
1589: ′(
1569:odd, and
1496:−
1463:δ
1430:→
1415:→
1380:−
1347:∇
1318:→
1303:→
1268:−
1235:Δ
1160:Δ
1132:−
1064:−
1035:→
923:∇
884:Δ
862:−
850:−
793:δ
753:−
729:Δ
716:−
704:−
662:∇
649:and
588:−
540:Δ
515:Δ
470:−
422:Δ
376:Δ
252:involves
213:−
171:Δ
142:Δ
114:Δ
89:numerical
13982:Manifold
13715:Integral
13658:Infinite
13653:Harmonic
13638:Binomial
13484:Gradient
13427:Volumes
13238:Quotient
13179:Notation
13010:Calculus
12878:(1995).
12859:(1968).
12793:14106142
12734:16797959
12642:(1872).
12543:: 1–45.
12349:(2013).
12231:See also
12209: (
12186: (
10330: .
10174:constant
9592: (
9301: .
9267: (
9007: ′(
8852: ,
8812: .
8810: ′
8383: .
8322: .
8303: .
8256: ,
8246: (
8212: T
8136:operator
7941:and the
6767:is the "
6268:ordinary
3327:sequence
2800:Backward
2365:″
2085:″
1812:″
1784: ′
1574: (
1559: (
1503:′
1423:as
1387:′
1311:as
1275:′
1011:′
408:function
350:backward
314:calculus
282:(1860),
128:operator
87:for the
53: (
41: (
13919:inverse
13907:inverse
13833:Fluxion
13643:Fourier
13509:Stokes'
13504:Green's
13226:Product
13086:Tangent
12956:, 2001
12771:Bibcode
12714:Bibcode
12588:Bibcode
11195:over a
10326:
10176:, then
10001:of the
9803:
9784:
9279:
8839:
8805:
8341:
8307:
8271:
8262:is the
8230:
8220:is the
8218:
6670:is the
6289:, etc.
4687:, with
3156:, have
2943:Central
2643:Forward
1775:
1761:
1746:
1732:
1683:scholar
354:central
346:forward
126:is the
14002:Tensor
13924:Secant
13690:Abel's
13673:Taylor
13564:Matrix
13514:Gauss'
13096:Limits
13076:Secant
13066:Radian
12928:
12839:
12814:
12791:
12759:: 15.
12732:
12674:
12608:
12411:
12387:
12357:
12324:
12250:(FDTD)
11085:where
10945:
10786:
10755:
10512:
10481:
10352:
10301:
10292:
10277:
10262:
10250:
10238:
10199:
10184:
10119:
10023:
9975:
9817:
9763:
9626:
9609:
9572:
9547:
9505:
9482:
9469:
9355:
9349:
9321:
9128:
9089:arsinh
8980:
8795:where
8780:
8615:
8605:
8575:
8565:
8550:
8544:
8523:
8513:
8483:
8473:
8467:
8457:
8442:
8427:
8402:
8210:where
8195:
6674:, and
5487:Δ
5300:Δ
5292:Δ
5083:Δ
5075:Δ
5067:Δ
4956:(0, 0)
4632:l.o.t.
4593:l.o.t.
4541:l.o.t.
4300:where
3981:After
3888:l.o.t.
3308:
3302:
3292:
3286:
3273:
3248:
3228:
3222:
3212:
3206:
3193:
3168:
1685:
1678:
1671:
1664:
1656:
352:, and
13866:Lists
13725:Ratio
13663:Power
13399:Euler
13216:Chain
13206:Power
13081:Slope
12883:(PDF)
12789:S2CID
12761:arXiv
12730:S2CID
12704:arXiv
12606:S2CID
12578:arXiv
12545:arXiv
12533:(PDF)
12476:notes
11097:, …,
10218:: If
10172:is a
10168:: If
9313:),
8837:Thus
8335:is a
5375:−306
5372:−1065
5359:−306
5343:−306
5327:−147
4948:+ 1,
4930:6364
4922:2641
3240:and
1690:JSTOR
1676:books
1593:) = 0
1578:) = 2
1563:) = 1
998:limit
406:of a
295:]
244:is a
13735:Term
13730:Root
13469:Curl
12926:ISBN
12837:ISBN
12812:ISBN
12672:ISBN
12463:2017
12409:ISBN
12385:ISBN
12355:ISBN
12322:ISBN
12204:and
10324:and
10226:are
10222:and
8367:) +
8353:) =
8258:and
8242:) =
7658:>
6807:,
6319:and
6297:The
6270:and
5813:and
5626:= 36
5553:108
5540:108
5527:108
5514:108
5356:−759
5340:−453
5154:648
5151:1854
5148:3723
5145:6364
5137:648
5134:1206
5131:1869
5128:2641
5120:558
5106:105
4952:+ 1)
4914:772
4906:109
4419:Let
4282:Let
4091:Let
3881:and
3875:and
1750:and
1662:news
1580:for
1565:for
984:The
629:and
102:The
49:) −
13211:Sum
12895:doi
12891:144
12779:doi
12722:doi
12596:doi
12541:18A
12452:doi
12318:137
11090:= (
10668:or
10612:det
10547:det
10044:sin
9612:sin
9158:→ 0
8828:all
8284:≡ Δ
8146:to
7674:max
6949:),
6794:= 1
6555:is
5712:−19
5699:13
5689:10
5567:108
5545:13
5532:10
5458:−17
5454:−17
5392:306
5364:13
5348:10
5241:162
5183:648
5169:= 3
5162:648
5142:13
5125:10
5117:663
5114:772
5103:109
4698:− 1
4690:ahm
4685:− 1
4665:ahm
4662:As
3341:or
1786:at
1645:by
1099:If
1028:lim
316:of
263:In
14051::
12952:,
12946:,
12889:.
12885:.
12787:.
12777:.
12769:.
12755:.
12751:.
12728:.
12720:.
12712:.
12700:23
12698:.
12604:.
12594:.
12586:.
12574:10
12572:.
12568:.
12539:.
12535:.
12448:43
12446:.
12442:.
12369:^
12336:^
12320:.
12298:^
12227:.
12221:−
12217:,
12213:−
12198:+
12194:,
12190:+
11165:=
11160::
11135:=
11127::
10973:A
10774::
10504::
10340::
10230:,
10142:.
10005:.
9939:ln
9892:ln
9042:ln
9019:.
8871:ln
8847:=
8391::
8320:≡
8290:(Δ
8266:.
8250:+
7916:,
7849:,
7495:,
6942:.
6790:,
6285:,
6278:.
6070::
5763:19
5754:36
5738:17
5679:7
5669:4
5659:1
5631:36
5519:7
5506:4
5496:1
5463:.
5432:18
5332:7
5319:4
5309:1
5272:.
5223:27
5111:7
5100:4
5095:4
5092:1
4958::
4927:13
4919:10
4898:4
4860:,
4321::
4106::
4078:.
3891::
3349:.
3149:.
1794::
1759:−
1730:+
1607:.
1591:nh
1576:nh
1561:nh
1433:0.
1321:0.
981:.
653::
644:+
634:−
617:A
359:A
348:,
320:.
303:c.
293:de
240:A
99:.
68:−
57:+
45:+
32:A
13002:e
12995:t
12988:v
12901:.
12897::
12845:.
12820:.
12795:.
12781::
12773::
12763::
12757:1
12736:.
12724::
12716::
12706::
12652:.
12612:.
12598::
12590::
12580::
12553:.
12547::
12465:.
12454::
12393:.
12363:.
12330:.
12225:)
12223:k
12219:y
12215:h
12211:x
12207:f
12202:)
12200:k
12196:y
12192:h
12188:x
12184:f
12169:,
12163:k
12160:h
12157:2
12152:)
12149:k
12143:y
12140:,
12137:h
12131:x
12128:(
12125:f
12122:+
12119:)
12116:k
12110:y
12107:,
12104:x
12101:(
12098:f
12092:)
12089:y
12086:,
12083:h
12077:x
12074:(
12071:f
12065:)
12062:y
12059:,
12056:x
12053:(
12050:f
12047:2
12044:+
12041:)
12038:k
12035:+
12032:y
12029:,
12026:x
12023:(
12020:f
12014:)
12011:y
12008:,
12005:h
12002:+
11999:x
11996:(
11993:f
11987:)
11984:k
11981:+
11978:y
11975:,
11972:h
11969:+
11966:x
11963:(
11960:f
11951:)
11948:y
11945:,
11942:x
11939:(
11934:y
11931:x
11927:f
11916:f
11896:.
11890:k
11887:h
11884:4
11879:)
11876:k
11870:y
11867:,
11864:h
11858:x
11855:(
11852:f
11849:+
11846:)
11843:k
11840:+
11837:y
11834:,
11831:h
11825:x
11822:(
11819:f
11813:)
11810:k
11804:y
11801:,
11798:h
11795:+
11792:x
11789:(
11786:f
11780:)
11777:k
11774:+
11771:y
11768:,
11765:h
11762:+
11759:x
11756:(
11753:f
11740:)
11737:y
11734:,
11731:x
11728:(
11723:y
11720:x
11716:f
11704:2
11700:k
11695:)
11692:k
11686:y
11683:,
11680:x
11677:(
11674:f
11671:+
11668:)
11665:y
11662:,
11659:x
11656:(
11653:f
11650:2
11644:)
11641:k
11638:+
11635:y
11632:,
11629:x
11626:(
11623:f
11610:)
11607:y
11604:,
11601:x
11598:(
11593:y
11590:y
11586:f
11574:2
11570:h
11565:)
11562:y
11559:,
11556:h
11550:x
11547:(
11544:f
11541:+
11538:)
11535:y
11532:,
11529:x
11526:(
11523:f
11520:2
11514:)
11511:y
11508:,
11505:h
11502:+
11499:x
11496:(
11493:f
11480:)
11477:y
11474:,
11471:x
11468:(
11463:x
11460:x
11456:f
11445:k
11442:2
11437:)
11434:k
11428:y
11425:,
11422:x
11419:(
11416:f
11410:)
11407:k
11404:+
11401:y
11398:,
11395:x
11392:(
11389:f
11376:)
11373:y
11370:,
11367:x
11364:(
11359:y
11355:f
11344:h
11341:2
11336:)
11333:y
11330:,
11327:h
11321:x
11318:(
11315:f
11309:)
11306:y
11303:,
11300:h
11297:+
11294:x
11291:(
11288:f
11275:)
11272:y
11269:,
11266:x
11263:(
11258:x
11254:f
11225:.
11219:μ
11215:μ
11199:.
11185:R
11179:.
11173:)
11171:x
11169:(
11167:h
11163:h
11158:x
11154:h
11146:)
11144:x
11142:(
11139:k
11137:μ
11132:k
11130:μ
11125:x
11119:k
11117:μ
11104:)
11101:N
11099:μ
11095:0
11092:μ
11088:μ
11073:,
11070:)
11067:h
11064:k
11061:+
11058:x
11055:(
11052:f
11047:k
11037:N
11032:0
11029:=
11026:k
11018:=
11015:)
11012:x
11009:(
11006:]
11003:f
11000:[
10990:h
10942:)
10939:1
10933:a
10930:(
10927:f
10921:)
10918:b
10915:(
10912:f
10909:=
10902:)
10899:n
10896:(
10893:f
10885:b
10880:a
10877:=
10874:n
10862:)
10859:a
10856:(
10853:f
10847:)
10844:1
10841:+
10838:b
10835:(
10832:f
10829:=
10822:)
10819:n
10816:(
10813:f
10805:b
10800:a
10797:=
10794:n
10749:)
10746:g
10737:g
10734:(
10728:g
10723:g
10716:f
10710:f
10703:g
10697:=
10693:)
10688:g
10685:f
10680:(
10655:)
10649:]
10643:1
10638:1
10631:g
10623:g
10617:[
10606:g
10602:(
10597:/
10593:)
10587:]
10581:g
10576:f
10569:g
10561:f
10552:[
10543:(
10536:=
10532:)
10527:g
10524:f
10519:(
10478:g
10471:f
10462:f
10455:g
10452:+
10449:g
10442:f
10439:=
10432:)
10429:g
10426:f
10423:(
10413:g
10406:f
10400:+
10397:f
10390:g
10387:+
10384:g
10377:f
10374:=
10367:)
10364:g
10361:f
10358:(
10328:∇
10322:δ
10318:Δ
10298:g
10289:b
10286:+
10283:f
10274:a
10271:=
10268:)
10265:g
10259:b
10256:+
10253:f
10247:a
10244:(
10224:b
10220:a
10196:0
10193:=
10190:c
10170:c
10122:,
10113:)
10110:h
10107:+
10104:x
10101:(
10092:]
10087:)
10081:h
10078:x
10073:+
10070:1
10066:(
10060:2
10051:[
10035:)
10032:x
10029:(
9978:,
9968:h
9965:x
9960:)
9957:h
9951:+
9948:1
9945:(
9935:e
9928:=
9921:h
9918:x
9913:)
9910:h
9904:+
9901:1
9898:(
9888:e
9882:h
9877:h
9867:=
9861:h
9858:x
9853:)
9849:h
9843:+
9840:1
9837:(
9832:h
9827:h
9799:h
9795:/
9790:h
9787:Δ
9757:+
9751:!
9748:7
9741:7
9737:)
9733:x
9730:(
9718:!
9715:5
9708:5
9704:)
9700:x
9697:(
9691:+
9685:!
9682:3
9675:3
9671:)
9667:x
9664:(
9655:x
9652:=
9648:)
9642:1
9634:h
9630:T
9623:x
9619:(
9596:)
9594:x
9590:f
9575:,
9567:1
9561:n
9557:)
9553:x
9550:(
9544:n
9541:=
9536:n
9532:)
9528:x
9525:(
9520:h
9515:h
9485:,
9477:)
9472:h
9465:)
9461:1
9455:n
9451:(
9444:x
9439:(
9430:)
9426:h
9423:2
9417:x
9413:(
9408:)
9404:h
9398:x
9394:(
9390:x
9387:=
9382:n
9377:)
9371:1
9363:h
9359:T
9352:x
9345:(
9335:n
9331:)
9327:x
9324:(
9307:x
9299:)
9294:h
9288:T
9286:x
9282:f
9271:)
9269:x
9265:f
9247:.
9244:I
9241:=
9238:]
9235:x
9232:,
9229:D
9226:[
9223:=
9219:]
9213:1
9205:h
9201:T
9196:x
9193:,
9188:h
9183:h
9172:[
9156:h
9131:.
9124:)
9118:h
9106:2
9103:1
9096:(
9086:2
9083:=
9080:D
9077:h
9067:)
9062:h
9051:1
9048:(
9036:=
9033:D
9030:h
9011:)
9009:x
9005:f
8983:.
8969:3
8964:h
8952:3
8949:1
8943:+
8938:2
8933:h
8921:2
8918:1
8907:h
8899:=
8896:)
8891:h
8883:+
8880:1
8877:(
8868:=
8865:D
8862:h
8849:e
8844:h
8841:T
8835:.
8833:h
8820:h
8808:f
8801:f
8797:D
8783:,
8774:I
8766:D
8763:h
8759:e
8755:=
8749:+
8744:3
8740:D
8734:3
8730:h
8723:!
8720:3
8716:1
8711:+
8706:2
8702:D
8696:2
8692:h
8685:!
8682:2
8678:1
8673:+
8670:D
8667:h
8664:=
8659:h
8642:h
8618:.
8610:)
8602:)
8599:x
8596:(
8593:g
8585:h
8570:(
8562:)
8559:x
8556:(
8553:f
8547:+
8541:)
8538:h
8535:+
8532:x
8529:(
8526:g
8518:)
8510:)
8507:x
8504:(
8501:f
8493:h
8478:(
8470:=
8462:)
8454:)
8451:x
8448:(
8445:g
8439:)
8436:x
8433:(
8430:f
8422:(
8412:h
8381:)
8379:x
8377:(
8374:h
8371:Δ
8369:β
8365:x
8363:(
8360:h
8357:Δ
8355:α
8351:x
8349:(
8346:h
8343:Δ
8331:h
8328:Δ
8315:h
8309:Δ
8301:)
8296:h
8287:h
8279:h
8273:Δ
8254:)
8252:h
8248:x
8244:f
8240:x
8238:(
8235:h
8232:T
8226:h
8215:h
8198:,
8189:I
8181:h
8177:T
8173:=
8168:h
8151:h
8148:Δ
8144:f
8115:.
8112:)
8109:h
8106:j
8103:+
8100:a
8097:(
8094:f
8088:)
8083:j
8080:k
8075:(
8067:j
8061:k
8057:)
8053:1
8047:(
8042:k
8037:0
8034:=
8031:j
8020:)
8015:k
8011:h
8007:a
8001:x
7994:(
7986:0
7983:=
7980:k
7972:=
7969:)
7966:x
7963:(
7960:f
7926:f
7922:f
7912:p
7895:.
7891:)
7885:(
7881:P
7875:0
7869:=
7866:)
7860:(
7857:f
7833:,
7829:)
7823:k
7819:x
7808:(
7799:k
7795:P
7791:=
7786:1
7783:+
7780:k
7776:P
7770:,
7767:1
7764:=
7758:0
7754:P
7730:k
7727:,
7724:0
7716:=
7711:k
7707:0
7699:,
7695:}
7691:k
7684:)
7681:j
7678:(
7668:j
7664:,
7661:0
7655:k
7651:{
7637:j
7633:x
7624:k
7621:+
7618:j
7614:x
7606:1
7600:k
7597:,
7594:j
7581:1
7575:k
7572:,
7569:1
7566:+
7563:j
7552:=
7547:k
7544:,
7541:j
7532:,
7527:j
7523:y
7519:=
7514:0
7511:,
7508:j
7489:x
7465:)
7462:2
7456:x
7453:(
7450:)
7447:1
7441:x
7438:(
7435:+
7432:2
7429:=
7413:2
7409:)
7406:2
7400:x
7397:(
7394:)
7391:1
7385:x
7382:(
7372:2
7369:+
7363:1
7359:1
7353:x
7343:0
7340:+
7337:1
7331:2
7328:=
7315:)
7312:1
7309:=
7304:0
7300:x
7296:(
7288:!
7285:2
7278:2
7274:)
7268:0
7264:x
7257:x
7254:(
7242:2
7234:+
7227:!
7224:1
7217:1
7213:)
7207:0
7203:x
7196:x
7193:(
7181:1
7173:+
7170:1
7162:0
7154:=
7147:)
7144:x
7141:(
7138:f
7122:4
7117:3
7109:2
7095:2
7087:2
7082:2
7069:0
7051:2
7044:1
7035:2
7023:1
7011:0
7003:=
7000:f
6995:x
6976:0
6973:x
6962:f
6922:,
6917:k
6913:)
6909:y
6906:(
6900:k
6894:n
6890:)
6886:x
6883:(
6877:)
6872:k
6869:n
6864:(
6856:n
6851:0
6848:=
6845:k
6837:=
6832:n
6828:)
6824:y
6821:+
6818:x
6815:(
6792:h
6788:x
6782:0
6780:)
6778:x
6776:(
6755:)
6752:1
6749:+
6746:k
6740:x
6737:(
6731:)
6728:2
6722:x
6719:(
6716:)
6713:1
6707:x
6704:(
6701:x
6698:=
6693:k
6689:)
6685:x
6682:(
6655:!
6652:k
6645:k
6641:)
6637:x
6634:(
6628:=
6622:)
6617:k
6614:x
6609:(
6553:f
6545:f
6524:,
6521:)
6518:a
6515:(
6512:]
6509:f
6506:[
6501:k
6489:)
6484:k
6480:a
6474:x
6468:(
6455:0
6452:=
6449:k
6441:=
6436:k
6432:)
6428:a
6422:x
6419:(
6412:!
6409:k
6404:)
6401:a
6398:(
6395:]
6392:f
6389:[
6384:k
6366:0
6363:=
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59:a
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39:f
20:)
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