1293:
2603:
4017:
3476:
of the disk of convergence. A power series may diverge at every point on the boundary, or diverge on some points and converge at other points, or converge at all the points on the boundary. Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily
79:
to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the
4236:
1282:
551:
The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms. In this second case, extrapolating a plot estimates the radius of
3812:
735:
1505:
increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity. In this case, two main techniques have been developed, based on the fact that the coefficients of a Taylor series are roughly exponential with ratio
481:
2089:
1374:
2450:
2973:
1098:
3085:
949:
4054:
1109:
840:
4376:
3753:
4012:{\displaystyle \sum _{i=1}^{\infty }a_{i}z^{i}{\text{ where }}a_{i}={\frac {(-1)^{n-1}}{2^{n}n}}{\text{ for }}n=\lfloor \log _{2}(i)\rfloor +1{\text{, the unique integer with }}2^{n-1}\leq i<2^{n},}
3305:
3646:
181:
2829:
1686:
1920:
571:
1606:
3103:. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem of complex analysis stated above quickly solves the problem. At
355:
3554:
2717:
2646:
1917:
The more complicated case is when the signs of the coefficients have a more complex pattern. Mercer and
Roberts proposed the following procedure. Define the associated sequence
1419:
1754:
2552:
2283:
1299:
344:
293:
3155:
2331:
1842:
2326:
4263:
2753:
248:
61:
2526:
1453:
2871:
2212:
2178:
2116:
1880:
1816:
1483:
546:
2485:
2240:
2144:
1908:
1782:
1532:
2260:
1706:
1503:
960:
4291:
So for these particular values the fastest convergence of a power series expansion is at the center, and as one moves away from the center of convergence, the
1542:
The basic case is when the coefficients ultimately share a common sign or alternate in sign. As pointed out earlier in the article, in many cases the limit
4269:. Both the number of terms and the value at which the series is to be evaluated affect the accuracy of the answer. For example, if we want to calculate
2992:
4231:{\displaystyle \sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots {\text{ for all }}x}
874:
2242:. This procedure also estimates two other characteristics of the convergence limiting singularity. Suppose the nearest singularity is of degree
1277:{\displaystyle |z-a|<{\frac {1}{\lim _{n\to \infty }{\frac {|c_{n+1}|}{|c_{n}|}}}}=\lim _{n\to \infty }\left|{\frac {c_{n}}{c_{n+1}}}\right|.}
766:
4498:
Mercer, G.N.; Roberts, A.J. (1990), "A centre manifold description of contaminant dispersion in channels with varying flow properties",
4265:
meaning that this series converges for all complex numbers. However, in applications, one is often interested in the precision of a
4316:
3693:
4273:
accurate up to five decimal places, we only need the first two terms of the series. However, if we want the same precision for
3199:
3590:
98:
1384:
in the Domb–Sykes plot, plot (b), which intercepts the vertical axis at −2 and has a slope +1. Thus there is a singularity at
4567:
730:{\displaystyle C=\limsup _{n\to \infty }{\sqrt{|c_{n}(z-a)^{n}|}}=\limsup _{n\to \infty }\left({\sqrt{|c_{n}|}}\right)|z-a|}
2850:
2761:
2567:
by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem:
1611:
4649:
4592:
4441:
1545:
476:{\displaystyle r=\sup \left\{|z-a|\ \left|\ \sum _{n=0}^{\infty }c_{n}(z-a)^{n}\ {\text{ converges }}\right.\right\}}
3508:
4629:
2659:, not necessarily on the real line, even if the center and all coefficients are real. For example, the function
2665:
4654:
4415:
868:
is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite.
2084:{\displaystyle b_{n}^{2}={\frac {c_{n+1}c_{n-1}-c_{n}^{2}}{c_{n}c_{n-2}-c_{n-1}^{2}}}\quad n=3,4,5,\ldots .}
4529:"O szeregu potęgowym, który jest zbieżny na całem swem kole zbieżności jednostajnie, ale nie bezwzględnie"
2609:
4644:
1708:
means the convergence-limiting singularity is on the negative axis. Estimate this limit, by plotting the
846:
4584:
1387:
498:, the behavior of the power series may be complicated, and the series may converge for some values of
1369:{\displaystyle f(\varepsilon )={\frac {\varepsilon (1+\varepsilon ^{3})}{\sqrt {1+2\varepsilon }}}.}
2445:{\textstyle {\frac {1}{2}}\left({\frac {c_{n-1}b_{n}}{c_{n}}}+{\frac {c_{n+1}}{c_{n}b_{n}}}\right)}
1711:
2531:
4308:
2265:
304:
4456:
Domb, C.; Sykes, M.F. (1957), "On the susceptibility of a ferromagnetic above the Curie point",
4436:, Cambridge Texts in Applied Mathematics, vol. 6, Cambridge University Press, p. 146,
256:
4524:
3121:
2834:
The root test shows that its radius of convergence is 1. In accordance with this, the function
1821:
2968:{\displaystyle \arctan(z)=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots .}
2288:
502:
and diverge for others. The radius of convergence is infinite if the series converges for all
4248:
3108:
2725:
233:
46:
2978:
It is easy to apply the root test in this case to find that the radius of convergence is 1.
1424:
4465:
4296:
4037:
2564:
2490:
2183:
2149:
2094:
1851:
1787:
1461:
524:
64:
1093:{\displaystyle \lim _{n\to \infty }{\frac {|c_{n+1}(z-a)^{n+1}|}{|c_{n}(z-a)^{n}|}}<1.}
8:
4295:
slows down until you reach the boundary (if it exists) and cross over, in which case the
4292:
4023:
3412:
The singularities nearest 0, which is the center of the power series expansion, are at ±2
2455:
2217:
2121:
1885:
1759:
1509:
68:
4469:
4481:
4266:
2245:
1691:
1488:
198:
32:
4612:
4607:
4588:
4563:
4485:
4437:
3764:
is the function represented by this series on the unit disk, then the derivative of
3100:
1845:
76:
40:
3758:
has radius of convergence 1 and converges everywhere on the boundary absolutely. If
752: > 1. It follows that the power series converges if the distance from
4507:
4473:
80:
center of the disk of convergence to the respective singularities of the function.
3080:{\displaystyle {\frac {z}{e^{z}-1}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}z^{n}}
1287:
858:
71:
inside the open disk of radius equal to the radius of convergence, and it is the
4659:
4576:
741:
503:
194:
4638:
2656:
1378:
72:
944:{\displaystyle r=\lim _{n\to \infty }\left|{\frac {c_{n}}{c_{n+1}}}\right|.}
4477:
2862:
36:
28:
1458:
Usually, in scientific applications, only a finite number of coefficients
4559:
3800:
3559:
has radius of convergence 1 and diverges at every point on the boundary.
20:
2563:
A power series with a positive radius of convergence can be made into a
1292:
845:
and diverges if the distance exceeds that number; this statement is the
548:
then you take certain limits and find the precise radius of convergence.
3670:
954:
This is shown as follows. The ratio test says the series converges if
865:
835:{\displaystyle r={\frac {1}{\limsup _{n\to \infty }{\sqrt{|c_{n}|}}}}}
4617:
3111:. The only non-removable singularities are therefore located at the
1381:
562:
4511:
2602:
853: = 1/0 is interpreted as an infinite radius, meaning that
349:
Some may prefer an alternative definition, as existence is obvious:
4395:
2606:
Radius of convergence (white) and Taylor approximations (blue) for
2285:
to the real axis. Then the slope of the linear fit given above is
4385:
is greater than a particular number depending on the coefficients
4280:
we must evaluate and sum the first five terms of the series. For
4245:= 0, we find out that the radius of convergence of this series is
521:
The first case is theoretical: when you know all the coefficients
3658:
but converges for all other points on the boundary. The function
4302:
4528:
3469:
1288:
Practical estimation of radius in the case of real coefficients
3419:. The distance from the center to either of those points is 2
3394:. Consequently the singular points of this function occur at
2595:
is strictly less than the radius of convergence is called the
4371:{\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}
3748:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}z^{n}}
2558:
465:
3300:{\displaystyle e^{z}=e^{x}e^{iy}=e^{x}(\cos(y)+i\sin(y)),}
565:
to the terms of the series. The root test uses the number
4284:, one requires the first 18 terms of the series, and for
3641:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}z^{n},}
2755:
has no real roots. Its Taylor series about 0 is given by
176:{\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n},}
2214:
estimates the reciprocal of the radius of convergence,
1882:
estimates the reciprocal of the radius of convergence,
561:
The radius of convergence can be found by applying the
2493:
2458:
2334:
1614:
1548:
4319:
4251:
4057:
3815:
3696:
3593:
3511:
3202:
3124:
2995:
2874:
2764:
2728:
2668:
2612:
2587:
cannot be defined in a way that makes it holomorphic.
2534:
2291:
2268:
2248:
2220:
2186:
2152:
2124:
2097:
1923:
1888:
1854:
1824:
1790:
1762:
1714:
1694:
1512:
1491:
1464:
1427:
1390:
1302:
1112:
963:
877:
769:
744:. The root test states that the series converges if
574:
527:
358:
307:
259:
236:
101:
49:
3338:
is necessarily 1. Therefore, the absolute value of
2824:{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}z^{2n}.}
1681:{\textstyle 1/r=\lim _{n\to \infty }{c_{n}/c_{n-1}}}
512:
4370:
4257:
4230:
4011:
3747:
3640:
3548:
3299:
3149:
3079:
2967:
2823:
2747:
2711:
2640:
2546:
2520:
2479:
2444:
2320:
2277:
2254:
2234:
2206:
2172:
2138:
2110:
2083:
1902:
1874:
1836:
1810:
1776:
1748:
1700:
1680:
1600:
1526:
1497:
1477:
1447:
1413:
1368:
1276:
1092:
943:
834:
729:
540:
475:
338:
287:
242:
175:
55:
3435:If the power series is expanded around the point
4636:
3115:points where the denominator is zero. We solve
1630:
1601:{\textstyle \lim _{n\to \infty }{c_{n}/c_{n-1}}}
1550:
1218:
1142:
965:
885:
783:
654:
582:
365:
4575:
4553:
3651:has radius of convergence 1, and diverges for
3430:
43:. It is either a non-negative real number or
4497:
4309:abscissa of convergence of a Dirichlet series
4303:Abscissa of convergence of a Dirichlet series
3480:Example 1: The power series for the function
3107:= 0, there is in effect no singularity since
2981:
2722:has no singularities on the real line, since
4420:. Krishna Prakashan Media. 16 November 2010.
4381:Such a series converges if the real part of
3957:
3932:
2571:The radius of convergence of a power series
3549:{\displaystyle \sum _{n=0}^{\infty }z^{n},}
4523:
4022:has radius of convergence 1 and converges
4455:
4288:we need to evaluate the first 141 terms.
2846:, which are at a distance 1 from 0.
2712:{\displaystyle f(z)={\frac {1}{1+z^{2}}}}
2559:Radius of convergence in complex analysis
63:. When it is positive, the power series
3423:, so the radius of convergence is 2
2601:
2591:The set of all points whose distance to
1291:
748: < 1 and diverges if
4637:
4554:Brown, James; Churchill, Ruel (1989),
4043:
3401:= a nonzero integer multiple of 2
2180:via a linear fit. The intercept with
4431:
556:
16:Domain of convergence of power series
2851:analyticity of holomorphic functions
2641:{\displaystyle {\frac {1}{1+z^{2}}}}
2487:, then a linear fit extrapolated to
1421:and so the radius of convergence is
2865:can be expanded in a power series:
2856:
13:
4556:Complex variables and applications
4336:
4252:
4086:
3832:
3713:
3610:
3528:
3378:is real, that happens only if cos(
3040:
2781:
1831:
1640:
1560:
1228:
1152:
975:
895:
793:
664:
592:
419:
250:such that the series converges if
237:
133:
50:
14:
4671:
4623:
3439:and the radius of convergence is
2849:For a proof of this theorem, see
2146:, and graphically extrapolate to
1784:, and graphically extrapolate to
1414:{\displaystyle \varepsilon =-1/2}
513:Finding the radius of convergence
486:On the boundary, that is, where |
3788:of Example 2. It turns out that
3562:Example 2: The power series for
3390:is an integer multiple of 2
230:is a nonnegative real number or
3968:, the unique integer with
3322:is real, the absolute value of
2655:means the nearest point in the
2050:
4630:What is radius of convergence?
4517:
4491:
4449:
4424:
4408:
4130:
4115:
4104:
4094:
3954:
3948:
3888:
3878:
3350:is real, that happens only if
3291:
3288:
3282:
3267:
3261:
3252:
2887:
2881:
2796:
2786:
2678:
2672:
2579:is equal to the distance from
2307:
2295:
1637:
1557:
1538:is the radius of convergence.
1343:
1324:
1312:
1306:
1225:
1204:
1189:
1182:
1161:
1149:
1128:
1114:
1077:
1067:
1054:
1040:
1033:
1017:
1004:
984:
972:
892:
817:
802:
790:
723:
709:
692:
677:
661:
638:
628:
615:
601:
589:
447:
434:
388:
374:
323:
309:
275:
261:
161:
148:
111:
105:
1:
4547:
3445:, then the set of all points
2842:) has singularities at ±
2091:Plot the finitely many known
1749:{\displaystyle c_{n}/c_{n-1}}
83:
31:is the radius of the largest
4307:An analogous concept is the
3806:Example 4: The power series
3687:Example 3: The power series
3109:the singularity is removable
2986:Consider this power series:
2547:{\displaystyle \cos \theta }
1377:The solid green line is the
216:-th complex coefficient, and
197:constant, the center of the
7:
4601:
4533:Prace Matematyczno-Fizyczne
3431:Convergence on the boundary
3090:where the rational numbers
2861:The arctangent function of
2583:to the nearest point where
2278:{\displaystyle \pm \theta }
740:"lim sup" denotes the
339:{\displaystyle |z-a|>r.}
10:
4676:
4585:Princeton University Press
4579:; Shakarchi, Rami (2003),
4048:If we expand the function
2982:A more complicated example
1485:are known. Typically, as
864:The limit involved in the
288:{\displaystyle |z-a|<r}
226:The radius of convergence
4650:Convergence (mathematics)
4583:, Princeton, New Jersey:
3150:{\displaystyle e^{z}-1=0}
1910:. This plot is called a
1837:{\displaystyle n=\infty }
1608:exists, and in this case
69:uniformly on compact sets
4417:Mathematical Analysis-II
4401:
3358:is purely imaginary and
2321:{\displaystyle -(p+1)/r}
4258:{\displaystyle \infty }
4026:on the entire boundary
2748:{\displaystyle 1+z^{2}}
847:Cauchy–Hadamard theorem
243:{\displaystyle \infty }
56:{\displaystyle \infty }
4478:10.1098/rspa.1957.0078
4372:
4340:
4259:
4232:
4090:
4013:
3836:
3749:
3717:
3642:
3614:
3550:
3532:
3301:
3151:
3081:
3044:
2969:
2825:
2785:
2749:
2713:
2649:
2642:
2548:
2522:
2521:{\textstyle 1/n^{2}=0}
2481:
2446:
2322:
2279:
2256:
2236:
2208:
2174:
2140:
2112:
2085:
1904:
1876:
1848:. The intercept with
1838:
1812:
1778:
1750:
1702:
1682:
1602:
1528:
1499:
1479:
1455:
1449:
1448:{\displaystyle r=1/2.}
1415:
1370:
1296:Plots of the function
1278:
1103:That is equivalent to
1094:
945:
836:
731:
542:
477:
423:
340:
289:
244:
222:is a complex variable.
177:
137:
57:
4458:Proc. R. Soc. Lond. A
4373:
4320:
4260:
4233:
4070:
4014:
3816:
3750:
3697:
3643:
3594:
3551:
3512:
3477:converge absolutely.
3302:
3160:by recalling that if
3152:
3082:
3024:
2970:
2826:
2765:
2750:
2714:
2643:
2605:
2549:
2523:
2482:
2447:
2323:
2280:
2257:
2237:
2209:
2207:{\displaystyle 1/n=0}
2175:
2173:{\displaystyle 1/n=0}
2141:
2113:
2111:{\displaystyle b_{n}}
2086:
1905:
1877:
1875:{\displaystyle 1/n=0}
1839:
1813:
1811:{\displaystyle 1/n=0}
1779:
1751:
1703:
1683:
1603:
1529:
1500:
1480:
1478:{\displaystyle c_{n}}
1450:
1416:
1371:
1295:
1279:
1095:
946:
837:
732:
543:
541:{\displaystyle c_{n}}
478:
461: converges
403:
341:
290:
245:
178:
117:
58:
25:radius of convergence
4655:Mathematical physics
4434:Perturbation Methods
4432:Hinch, E.J. (1991),
4317:
4249:
4055:
3813:
3694:
3669:of Example 1 is the
3591:
3509:
3200:
3122:
2993:
2872:
2762:
2726:
2666:
2610:
2575:centered on a point
2565:holomorphic function
2532:
2491:
2480:{\textstyle 1/n^{2}}
2456:
2332:
2289:
2266:
2246:
2218:
2184:
2150:
2122:
2095:
1921:
1886:
1852:
1822:
1788:
1760:
1712:
1692:
1612:
1546:
1510:
1489:
1462:
1425:
1388:
1300:
1110:
961:
875:
767:
572:
525:
356:
305:
257:
234:
99:
65:converges absolutely
47:
39:in which the series
37:center of the series
4500:SIAM J. Appl. Math.
4470:1957RSPSA.240..214D
4430:See Figure 8.1 in:
4293:rate of convergence
4222: for all
4044:Rate of convergence
4038:converge absolutely
3318:to be real. Since
2597:disk of convergence
2235:{\displaystyle 1/r}
2139:{\displaystyle 1/n}
2046:
1994:
1938:
1903:{\displaystyle 1/r}
1777:{\displaystyle 1/n}
1527:{\displaystyle 1/r}
88:For a power series
4645:Analytic functions
4368:
4255:
4228:
4009:
3745:
3638:
3577:, expanded around
3546:
3502:, which is simply
3495:, expanded around
3297:
3147:
3077:
2965:
2821:
2745:
2709:
2650:
2638:
2544:
2518:
2477:
2442:
2318:
2275:
2252:
2232:
2204:
2170:
2136:
2108:
2081:
2026:
1980:
1924:
1900:
1872:
1834:
1808:
1774:
1746:
1698:
1678:
1644:
1598:
1564:
1524:
1495:
1475:
1456:
1445:
1411:
1366:
1274:
1232:
1156:
1090:
979:
941:
899:
832:
797:
727:
668:
596:
557:Theoretical radius
538:
473:
336:
285:
240:
173:
53:
4613:Convergence tests
4569:978-0-07-010905-6
4464:(1221): 214–228,
4363:
4241:around the point
4223:
4212:
4187:
4137:
4040:on the boundary.
3969:
3924:
3919:
3860:
3859: where
3733:
3623:
3342:can be 1 only if
3101:Bernoulli numbers
3065:
3019:
2954:
2934:
2914:
2707:
2653:The nearest point
2636:
2528:has intercept at
2435:
2390:
2343:
2328:. Further, plot
2255:{\displaystyle p}
2048:
1701:{\displaystyle r}
1629:
1549:
1498:{\displaystyle n}
1361:
1360:
1265:
1217:
1212:
1209:
1141:
1082:
964:
932:
884:
830:
827:
782:
702:
653:
648:
581:
517:Two cases arise:
462:
458:
402:
394:
77:analytic function
4667:
4597:
4581:Complex Analysis
4572:
4541:
4540:
4521:
4515:
4514:
4506:(6): 1547–1565,
4495:
4489:
4488:
4453:
4447:
4446:
4428:
4422:
4421:
4412:
4398:of convergence.
4377:
4375:
4374:
4369:
4364:
4362:
4361:
4352:
4351:
4342:
4339:
4334:
4287:
4283:
4279:
4272:
4267:numerical answer
4264:
4262:
4261:
4256:
4237:
4235:
4234:
4229:
4224:
4221:
4213:
4211:
4203:
4202:
4193:
4188:
4186:
4178:
4177:
4168:
4157:
4156:
4138:
4136:
4113:
4112:
4111:
4092:
4089:
4084:
4035:
4033:
4018:
4016:
4015:
4010:
4005:
4004:
3986:
3985:
3970:
3967:
3944:
3943:
3925:
3922:
3920:
3918:
3914:
3913:
3903:
3902:
3901:
3876:
3871:
3870:
3861:
3858:
3856:
3855:
3846:
3845:
3835:
3830:
3798:
3763:
3754:
3752:
3751:
3746:
3744:
3743:
3734:
3732:
3731:
3719:
3716:
3711:
3683:
3668:
3657:
3647:
3645:
3644:
3639:
3634:
3633:
3624:
3616:
3613:
3608:
3583:
3576:
3555:
3553:
3552:
3547:
3542:
3541:
3531:
3526:
3501:
3494:
3467:
3462:
3450:
3444:
3426:
3422:
3415:
3404:
3393:
3373:
3354:= 0. Therefore
3337:
3306:
3304:
3303:
3298:
3251:
3250:
3238:
3237:
3225:
3224:
3212:
3211:
3192:
3173:
3156:
3154:
3153:
3148:
3134:
3133:
3086:
3084:
3083:
3078:
3076:
3075:
3066:
3064:
3056:
3055:
3046:
3043:
3038:
3020:
3018:
3011:
3010:
2997:
2974:
2972:
2971:
2966:
2955:
2950:
2949:
2940:
2935:
2930:
2929:
2920:
2915:
2910:
2909:
2900:
2857:A simple example
2830:
2828:
2827:
2822:
2817:
2816:
2804:
2803:
2784:
2779:
2754:
2752:
2751:
2746:
2744:
2743:
2718:
2716:
2715:
2710:
2708:
2706:
2705:
2704:
2685:
2647:
2645:
2644:
2639:
2637:
2635:
2634:
2633:
2614:
2553:
2551:
2550:
2545:
2527:
2525:
2524:
2519:
2511:
2510:
2501:
2486:
2484:
2483:
2478:
2476:
2475:
2466:
2451:
2449:
2448:
2443:
2441:
2437:
2436:
2434:
2433:
2432:
2423:
2422:
2412:
2411:
2396:
2391:
2389:
2388:
2379:
2378:
2377:
2368:
2367:
2351:
2344:
2336:
2327:
2325:
2324:
2319:
2314:
2284:
2282:
2281:
2276:
2261:
2259:
2258:
2253:
2241:
2239:
2238:
2233:
2228:
2213:
2211:
2210:
2205:
2194:
2179:
2177:
2176:
2171:
2160:
2145:
2143:
2142:
2137:
2132:
2117:
2115:
2114:
2109:
2107:
2106:
2090:
2088:
2087:
2082:
2049:
2047:
2045:
2040:
2022:
2021:
2006:
2005:
1995:
1993:
1988:
1976:
1975:
1960:
1959:
1943:
1937:
1932:
1909:
1907:
1906:
1901:
1896:
1881:
1879:
1878:
1873:
1862:
1843:
1841:
1840:
1835:
1817:
1815:
1814:
1809:
1798:
1783:
1781:
1780:
1775:
1770:
1755:
1753:
1752:
1747:
1745:
1744:
1729:
1724:
1723:
1707:
1705:
1704:
1699:
1687:
1685:
1684:
1679:
1677:
1676:
1675:
1660:
1655:
1654:
1643:
1622:
1607:
1605:
1604:
1599:
1597:
1596:
1595:
1580:
1575:
1574:
1563:
1533:
1531:
1530:
1525:
1520:
1504:
1502:
1501:
1496:
1484:
1482:
1481:
1476:
1474:
1473:
1454:
1452:
1451:
1446:
1441:
1420:
1418:
1417:
1412:
1407:
1375:
1373:
1372:
1367:
1362:
1347:
1346:
1342:
1341:
1319:
1283:
1281:
1280:
1275:
1270:
1266:
1264:
1263:
1248:
1247:
1238:
1231:
1213:
1211:
1210:
1208:
1207:
1202:
1201:
1192:
1186:
1185:
1180:
1179:
1164:
1158:
1155:
1136:
1131:
1117:
1099:
1097:
1096:
1091:
1083:
1081:
1080:
1075:
1074:
1053:
1052:
1043:
1037:
1036:
1031:
1030:
1003:
1002:
987:
981:
978:
950:
948:
947:
942:
937:
933:
931:
930:
915:
914:
905:
898:
841:
839:
838:
833:
831:
829:
828:
826:
821:
820:
815:
814:
805:
799:
796:
777:
736:
734:
733:
728:
726:
712:
707:
703:
701:
696:
695:
690:
689:
680:
674:
667:
649:
647:
642:
641:
636:
635:
614:
613:
604:
598:
595:
547:
545:
544:
539:
537:
536:
482:
480:
479:
474:
472:
468:
467:
464:
463:
460:
456:
455:
454:
433:
432:
422:
417:
400:
392:
391:
377:
345:
343:
342:
337:
326:
312:
298:and diverges if
294:
292:
291:
286:
278:
264:
249:
247:
246:
241:
182:
180:
179:
174:
169:
168:
147:
146:
136:
131:
62:
60:
59:
54:
4675:
4674:
4670:
4669:
4668:
4666:
4665:
4664:
4635:
4634:
4626:
4604:
4595:
4570:
4550:
4545:
4544:
4522:
4518:
4512:10.1137/0150091
4496:
4492:
4454:
4450:
4444:
4429:
4425:
4414:
4413:
4409:
4404:
4393:
4357:
4353:
4347:
4343:
4341:
4335:
4324:
4318:
4315:
4314:
4305:
4285:
4281:
4274:
4270:
4250:
4247:
4246:
4220:
4204:
4198:
4194:
4192:
4179:
4173:
4169:
4167:
4143:
4139:
4114:
4107:
4103:
4093:
4091:
4085:
4074:
4056:
4053:
4052:
4046:
4036:, but does not
4029:
4027:
4000:
3996:
3975:
3971:
3966:
3939:
3935:
3923: for
3921:
3909:
3905:
3904:
3891:
3887:
3877:
3875:
3866:
3862:
3857:
3851:
3847:
3841:
3837:
3831:
3820:
3814:
3811:
3810:
3789:
3759:
3739:
3735:
3727:
3723:
3718:
3712:
3701:
3695:
3692:
3691:
3674:
3659:
3652:
3629:
3625:
3615:
3609:
3598:
3592:
3589:
3588:
3578:
3563:
3537:
3533:
3527:
3516:
3510:
3507:
3506:
3496:
3481:
3454:
3452:
3446:
3440:
3433:
3424:
3420:
3413:
3402:
3391:
3386:) = 0, so that
3359:
3323:
3246:
3242:
3230:
3226:
3220:
3216:
3207:
3203:
3201:
3198:
3197:
3175:
3161:
3129:
3125:
3123:
3120:
3119:
3098:
3071:
3067:
3057:
3051:
3047:
3045:
3039:
3028:
3006:
3002:
3001:
2996:
2994:
2991:
2990:
2984:
2945:
2941:
2939:
2925:
2921:
2919:
2905:
2901:
2899:
2873:
2870:
2869:
2859:
2809:
2805:
2799:
2795:
2780:
2769:
2763:
2760:
2759:
2739:
2735:
2727:
2724:
2723:
2700:
2696:
2689:
2684:
2667:
2664:
2663:
2629:
2625:
2618:
2613:
2611:
2608:
2607:
2561:
2533:
2530:
2529:
2506:
2502:
2497:
2492:
2489:
2488:
2471:
2467:
2462:
2457:
2454:
2453:
2428:
2424:
2418:
2414:
2413:
2401:
2397:
2395:
2384:
2380:
2373:
2369:
2357:
2353:
2352:
2350:
2349:
2345:
2335:
2333:
2330:
2329:
2310:
2290:
2287:
2286:
2267:
2264:
2263:
2247:
2244:
2243:
2224:
2219:
2216:
2215:
2190:
2185:
2182:
2181:
2156:
2151:
2148:
2147:
2128:
2123:
2120:
2119:
2102:
2098:
2096:
2093:
2092:
2041:
2030:
2011:
2007:
2001:
1997:
1996:
1989:
1984:
1965:
1961:
1949:
1945:
1944:
1942:
1933:
1928:
1922:
1919:
1918:
1912:Domb–Sykes plot
1892:
1887:
1884:
1883:
1858:
1853:
1850:
1849:
1823:
1820:
1819:
1794:
1789:
1786:
1785:
1766:
1761:
1758:
1757:
1734:
1730:
1725:
1719:
1715:
1713:
1710:
1709:
1693:
1690:
1689:
1665:
1661:
1656:
1650:
1646:
1645:
1633:
1618:
1613:
1610:
1609:
1585:
1581:
1576:
1570:
1566:
1565:
1553:
1547:
1544:
1543:
1516:
1511:
1508:
1507:
1490:
1487:
1486:
1469:
1465:
1463:
1460:
1459:
1437:
1426:
1423:
1422:
1403:
1389:
1386:
1385:
1376:
1337:
1333:
1320:
1318:
1301:
1298:
1297:
1290:
1253:
1249:
1243:
1239:
1237:
1233:
1221:
1203:
1197:
1193:
1188:
1187:
1181:
1169:
1165:
1160:
1159:
1157:
1145:
1140:
1135:
1127:
1113:
1111:
1108:
1107:
1076:
1070:
1066:
1048:
1044:
1039:
1038:
1032:
1020:
1016:
992:
988:
983:
982:
980:
968:
962:
959:
958:
920:
916:
910:
906:
904:
900:
888:
876:
873:
872:
859:entire function
822:
816:
810:
806:
801:
800:
798:
786:
781:
776:
768:
765:
764:
722:
708:
697:
691:
685:
681:
676:
675:
673:
669:
657:
643:
637:
631:
627:
609:
605:
600:
599:
597:
585:
573:
570:
569:
559:
532:
528:
526:
523:
522:
515:
504:complex numbers
459:
450:
446:
428:
424:
418:
407:
399:
395:
387:
373:
372:
368:
357:
354:
353:
322:
308:
306:
303:
302:
274:
260:
258:
255:
254:
235:
232:
231:
211:
201:of convergence,
164:
160:
142:
138:
132:
121:
100:
97:
96:
86:
48:
45:
44:
17:
12:
11:
5:
4673:
4663:
4662:
4657:
4652:
4647:
4633:
4632:
4625:
4624:External links
4622:
4621:
4620:
4615:
4610:
4608:Abel's theorem
4603:
4600:
4599:
4598:
4593:
4573:
4568:
4549:
4546:
4543:
4542:
4525:Sierpiński, W.
4516:
4490:
4448:
4442:
4423:
4406:
4405:
4403:
4400:
4389:
4379:
4378:
4367:
4360:
4356:
4350:
4346:
4338:
4333:
4330:
4327:
4323:
4304:
4301:
4299:will diverge.
4254:
4239:
4238:
4227:
4219:
4216:
4210:
4207:
4201:
4197:
4191:
4185:
4182:
4176:
4172:
4166:
4163:
4160:
4155:
4152:
4149:
4146:
4142:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4110:
4106:
4102:
4099:
4096:
4088:
4083:
4080:
4077:
4073:
4069:
4066:
4063:
4060:
4045:
4042:
4020:
4019:
4008:
4003:
3999:
3995:
3992:
3989:
3984:
3981:
3978:
3974:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3942:
3938:
3934:
3931:
3928:
3917:
3912:
3908:
3900:
3897:
3894:
3890:
3886:
3883:
3880:
3874:
3869:
3865:
3854:
3850:
3844:
3840:
3834:
3829:
3826:
3823:
3819:
3772:) is equal to
3756:
3755:
3742:
3738:
3730:
3726:
3722:
3715:
3710:
3707:
3704:
3700:
3649:
3648:
3637:
3632:
3628:
3622:
3619:
3612:
3607:
3604:
3601:
3597:
3557:
3556:
3545:
3540:
3536:
3530:
3525:
3522:
3519:
3515:
3432:
3429:
3410:
3409:
3382:) = 1 and sin(
3310:and then take
3308:
3307:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3249:
3245:
3241:
3236:
3233:
3229:
3223:
3219:
3215:
3210:
3206:
3158:
3157:
3146:
3143:
3140:
3137:
3132:
3128:
3094:
3088:
3087:
3074:
3070:
3063:
3060:
3054:
3050:
3042:
3037:
3034:
3031:
3027:
3023:
3017:
3014:
3009:
3005:
3000:
2983:
2980:
2976:
2975:
2964:
2961:
2958:
2953:
2948:
2944:
2938:
2933:
2928:
2924:
2918:
2913:
2908:
2904:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2858:
2855:
2832:
2831:
2820:
2815:
2812:
2808:
2802:
2798:
2794:
2791:
2788:
2783:
2778:
2775:
2772:
2768:
2742:
2738:
2734:
2731:
2720:
2719:
2703:
2699:
2695:
2692:
2688:
2683:
2680:
2677:
2674:
2671:
2632:
2628:
2624:
2621:
2617:
2589:
2588:
2560:
2557:
2556:
2555:
2543:
2540:
2537:
2517:
2514:
2509:
2505:
2500:
2496:
2474:
2470:
2465:
2461:
2440:
2431:
2427:
2421:
2417:
2410:
2407:
2404:
2400:
2394:
2387:
2383:
2376:
2372:
2366:
2363:
2360:
2356:
2348:
2342:
2339:
2317:
2313:
2309:
2306:
2303:
2300:
2297:
2294:
2274:
2271:
2262:and has angle
2251:
2231:
2227:
2223:
2203:
2200:
2197:
2193:
2189:
2169:
2166:
2163:
2159:
2155:
2135:
2131:
2127:
2105:
2101:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2044:
2039:
2036:
2033:
2029:
2025:
2020:
2017:
2014:
2010:
2004:
2000:
1992:
1987:
1983:
1979:
1974:
1971:
1968:
1964:
1958:
1955:
1952:
1948:
1941:
1936:
1931:
1927:
1915:
1899:
1895:
1891:
1871:
1868:
1865:
1861:
1857:
1833:
1830:
1827:
1807:
1804:
1801:
1797:
1793:
1773:
1769:
1765:
1743:
1740:
1737:
1733:
1728:
1722:
1718:
1697:
1674:
1671:
1668:
1664:
1659:
1653:
1649:
1642:
1639:
1636:
1632:
1628:
1625:
1621:
1617:
1594:
1591:
1588:
1584:
1579:
1573:
1569:
1562:
1559:
1556:
1552:
1523:
1519:
1515:
1494:
1472:
1468:
1444:
1440:
1436:
1433:
1430:
1410:
1406:
1402:
1399:
1396:
1393:
1365:
1359:
1356:
1353:
1350:
1345:
1340:
1336:
1332:
1329:
1326:
1323:
1317:
1314:
1311:
1308:
1305:
1289:
1286:
1285:
1284:
1273:
1269:
1262:
1259:
1256:
1252:
1246:
1242:
1236:
1230:
1227:
1224:
1220:
1216:
1206:
1200:
1196:
1191:
1184:
1178:
1175:
1172:
1168:
1163:
1154:
1151:
1148:
1144:
1139:
1134:
1130:
1126:
1123:
1120:
1116:
1101:
1100:
1089:
1086:
1079:
1073:
1069:
1065:
1062:
1059:
1056:
1051:
1047:
1042:
1035:
1029:
1026:
1023:
1019:
1015:
1012:
1009:
1006:
1001:
998:
995:
991:
986:
977:
974:
971:
967:
952:
951:
940:
936:
929:
926:
923:
919:
913:
909:
903:
897:
894:
891:
887:
883:
880:
843:
842:
825:
819:
813:
809:
804:
795:
792:
789:
785:
784:lim sup
780:
775:
772:
756:to the center
742:limit superior
738:
737:
725:
721:
718:
715:
711:
706:
700:
694:
688:
684:
679:
672:
666:
663:
660:
656:
655:lim sup
652:
646:
640:
634:
630:
626:
623:
620:
617:
612:
608:
603:
594:
591:
588:
584:
583:lim sup
580:
577:
558:
555:
554:
553:
549:
535:
531:
514:
511:
484:
483:
471:
466:
453:
449:
445:
442:
439:
436:
431:
427:
421:
416:
413:
410:
406:
398:
390:
386:
383:
380:
376:
371:
367:
364:
361:
347:
346:
335:
332:
329:
325:
321:
318:
315:
311:
296:
295:
284:
281:
277:
273:
270:
267:
263:
239:
224:
223:
217:
207:
202:
184:
183:
172:
167:
163:
159:
156:
153:
150:
145:
141:
135:
130:
127:
124:
120:
116:
113:
110:
107:
104:
85:
82:
52:
15:
9:
6:
4:
3:
2:
4672:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4642:
4640:
4631:
4628:
4627:
4619:
4616:
4614:
4611:
4609:
4606:
4605:
4596:
4594:0-691-11385-8
4590:
4586:
4582:
4578:
4574:
4571:
4565:
4561:
4557:
4552:
4551:
4539:(1): 263–266.
4538:
4534:
4530:
4526:
4520:
4513:
4509:
4505:
4501:
4494:
4487:
4483:
4479:
4475:
4471:
4467:
4463:
4459:
4452:
4445:
4443:0-521-37897-4
4439:
4435:
4427:
4419:
4418:
4411:
4407:
4399:
4397:
4392:
4388:
4384:
4365:
4358:
4354:
4348:
4344:
4331:
4328:
4325:
4321:
4313:
4312:
4311:
4310:
4300:
4298:
4294:
4289:
4277:
4268:
4244:
4225:
4217:
4214:
4208:
4205:
4199:
4195:
4189:
4183:
4180:
4174:
4170:
4164:
4161:
4158:
4153:
4150:
4147:
4144:
4140:
4133:
4127:
4124:
4121:
4118:
4108:
4100:
4097:
4081:
4078:
4075:
4071:
4067:
4064:
4061:
4058:
4051:
4050:
4049:
4041:
4039:
4032:
4025:
4006:
4001:
3997:
3993:
3990:
3987:
3982:
3979:
3976:
3972:
3963:
3960:
3951:
3945:
3940:
3936:
3929:
3926:
3915:
3910:
3906:
3898:
3895:
3892:
3884:
3881:
3872:
3867:
3863:
3852:
3848:
3842:
3838:
3827:
3824:
3821:
3817:
3809:
3808:
3807:
3804:
3802:
3796:
3792:
3787:
3783:
3779:
3775:
3771:
3767:
3762:
3740:
3736:
3728:
3724:
3720:
3708:
3705:
3702:
3698:
3690:
3689:
3688:
3685:
3681:
3677:
3672:
3666:
3662:
3655:
3635:
3630:
3626:
3620:
3617:
3605:
3602:
3599:
3595:
3587:
3586:
3585:
3581:
3574:
3570:
3566:
3560:
3543:
3538:
3534:
3523:
3520:
3517:
3513:
3505:
3504:
3503:
3499:
3492:
3488:
3484:
3478:
3475:
3471:
3466:
3461:
3457:
3449:
3443:
3438:
3428:
3418:
3407:
3400:
3397:
3396:
3395:
3389:
3385:
3381:
3377:
3371:
3367:
3363:
3357:
3353:
3349:
3345:
3341:
3335:
3331:
3327:
3321:
3317:
3313:
3294:
3285:
3279:
3276:
3273:
3270:
3264:
3258:
3255:
3247:
3243:
3239:
3234:
3231:
3227:
3221:
3217:
3213:
3208:
3204:
3196:
3195:
3194:
3190:
3186:
3182:
3178:
3172:
3168:
3164:
3144:
3141:
3138:
3135:
3130:
3126:
3118:
3117:
3116:
3114:
3110:
3106:
3102:
3097:
3093:
3072:
3068:
3061:
3058:
3052:
3048:
3035:
3032:
3029:
3025:
3021:
3015:
3012:
3007:
3003:
2998:
2989:
2988:
2987:
2979:
2962:
2959:
2956:
2951:
2946:
2942:
2936:
2931:
2926:
2922:
2916:
2911:
2906:
2902:
2896:
2893:
2890:
2884:
2878:
2875:
2868:
2867:
2866:
2864:
2854:
2852:
2847:
2845:
2841:
2837:
2818:
2813:
2810:
2806:
2800:
2792:
2789:
2776:
2773:
2770:
2766:
2758:
2757:
2756:
2740:
2736:
2732:
2729:
2701:
2697:
2693:
2690:
2686:
2681:
2675:
2669:
2662:
2661:
2660:
2658:
2657:complex plane
2654:
2630:
2626:
2622:
2619:
2615:
2604:
2600:
2598:
2594:
2586:
2582:
2578:
2574:
2570:
2569:
2568:
2566:
2541:
2538:
2535:
2515:
2512:
2507:
2503:
2498:
2494:
2472:
2468:
2463:
2459:
2438:
2429:
2425:
2419:
2415:
2408:
2405:
2402:
2398:
2392:
2385:
2381:
2374:
2370:
2364:
2361:
2358:
2354:
2346:
2340:
2337:
2315:
2311:
2304:
2301:
2298:
2292:
2272:
2269:
2249:
2229:
2225:
2221:
2201:
2198:
2195:
2191:
2187:
2167:
2164:
2161:
2157:
2153:
2133:
2129:
2125:
2103:
2099:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2042:
2037:
2034:
2031:
2027:
2023:
2018:
2015:
2012:
2008:
2002:
1998:
1990:
1985:
1981:
1977:
1972:
1969:
1966:
1962:
1956:
1953:
1950:
1946:
1939:
1934:
1929:
1925:
1916:
1913:
1897:
1893:
1889:
1869:
1866:
1863:
1859:
1855:
1847:
1828:
1825:
1818:(effectively
1805:
1802:
1799:
1795:
1791:
1771:
1767:
1763:
1741:
1738:
1735:
1731:
1726:
1720:
1716:
1695:
1672:
1669:
1666:
1662:
1657:
1651:
1647:
1634:
1626:
1623:
1619:
1615:
1592:
1589:
1586:
1582:
1577:
1571:
1567:
1554:
1541:
1540:
1539:
1537:
1521:
1517:
1513:
1492:
1470:
1466:
1442:
1438:
1434:
1431:
1428:
1408:
1404:
1400:
1397:
1394:
1391:
1383:
1380:
1379:straight-line
1363:
1357:
1354:
1351:
1348:
1338:
1334:
1330:
1327:
1321:
1315:
1309:
1303:
1294:
1271:
1267:
1260:
1257:
1254:
1250:
1244:
1240:
1234:
1222:
1214:
1198:
1194:
1176:
1173:
1170:
1166:
1146:
1137:
1132:
1124:
1121:
1118:
1106:
1105:
1104:
1087:
1084:
1071:
1063:
1060:
1057:
1049:
1045:
1027:
1024:
1021:
1013:
1010:
1007:
999:
996:
993:
989:
969:
957:
956:
955:
938:
934:
927:
924:
921:
917:
911:
907:
901:
889:
881:
878:
871:
870:
869:
867:
862:
860:
856:
852:
849:. Note that
848:
823:
811:
807:
787:
778:
773:
770:
763:
762:
761:
760:is less than
759:
755:
751:
747:
743:
719:
716:
713:
704:
698:
686:
682:
670:
658:
650:
644:
632:
624:
621:
618:
610:
606:
586:
578:
575:
568:
567:
566:
564:
550:
533:
529:
520:
519:
518:
510:
508:
505:
501:
497:
493:
490: −
489:
469:
451:
443:
440:
437:
429:
425:
414:
411:
408:
404:
396:
384:
381:
378:
369:
362:
359:
352:
351:
350:
333:
330:
327:
319:
316:
313:
301:
300:
299:
282:
279:
271:
268:
265:
253:
252:
251:
229:
221:
218:
215:
210:
206:
203:
200:
196:
192:
189:
188:
187:
170:
165:
157:
154:
151:
143:
139:
128:
125:
122:
118:
114:
108:
102:
95:
94:
93:
91:
81:
78:
74:
73:Taylor series
70:
66:
42:
38:
34:
30:
26:
22:
4580:
4577:Stein, Elias
4558:, New York:
4555:
4536:
4532:
4519:
4503:
4499:
4493:
4461:
4457:
4451:
4433:
4426:
4416:
4410:
4390:
4386:
4382:
4380:
4306:
4290:
4275:
4242:
4240:
4047:
4030:
4021:
3805:
3794:
3790:
3785:
3781:
3777:
3773:
3769:
3765:
3760:
3757:
3686:
3679:
3675:
3664:
3660:
3653:
3650:
3579:
3572:
3571:) = −ln(1 −
3568:
3564:
3561:
3558:
3497:
3490:
3486:
3482:
3479:
3473:
3464:
3459:
3455:
3447:
3441:
3436:
3434:
3416:
3411:
3405:
3398:
3387:
3383:
3379:
3375:
3369:
3365:
3361:
3355:
3351:
3347:
3346:is 1; since
3343:
3339:
3333:
3329:
3325:
3319:
3315:
3311:
3309:
3188:
3184:
3180:
3176:
3170:
3166:
3162:
3159:
3112:
3104:
3095:
3091:
3089:
2985:
2977:
2863:trigonometry
2860:
2848:
2843:
2839:
2835:
2833:
2721:
2652:
2651:
2596:
2592:
2590:
2584:
2580:
2576:
2572:
2562:
1911:
1535:
1457:
1102:
953:
863:
854:
850:
844:
757:
753:
749:
745:
739:
560:
552:convergence.
516:
506:
499:
495:
491:
487:
485:
348:
297:
227:
225:
219:
213:
208:
204:
190:
185:
92:defined as:
89:
87:
29:power series
24:
18:
4560:McGraw-Hill
3801:dilogarithm
3584:, which is
3489:) = 1/(1 −
3472:called the
1688:. Negative
21:mathematics
4639:Categories
4548:References
4034:| = 1
3803:function.
3671:derivative
3451:such that
1846:linear fit
866:ratio test
84:Definition
4618:Root test
4486:119974403
4337:∞
4322:∑
4253:∞
4218:⋯
4215:−
4165:−
4098:−
4087:∞
4072:∑
4062:
4024:uniformly
3988:≤
3980:−
3958:⌋
3946:
3933:⌊
3896:−
3882:−
3833:∞
3818:∑
3714:∞
3699:∑
3611:∞
3596:∑
3529:∞
3514:∑
3463:| =
3374:. Since
3280:
3259:
3136:−
3041:∞
3026:∑
3013:−
2960:⋯
2937:−
2897:−
2879:
2790:−
2782:∞
2767:∑
2542:θ
2539:
2362:−
2293:−
2273:θ
2270:±
2076:…
2035:−
2024:−
2016:−
1978:−
1970:−
1832:∞
1739:−
1670:−
1641:∞
1638:→
1590:−
1561:∞
1558:→
1398:−
1392:ε
1382:asymptote
1358:ε
1335:ε
1322:ε
1310:ε
1229:∞
1226:→
1153:∞
1150:→
1122:−
1061:−
1011:−
976:∞
973:→
896:∞
893:→
794:∞
791:→
717:−
665:∞
662:→
622:−
593:∞
590:→
563:root test
441:−
420:∞
405:∑
382:−
317:−
269:−
238:∞
155:−
134:∞
119:∑
51:∞
41:converges
4602:See also
4527:(1918).
4396:abscissa
4286:sin(100)
4271:sin(0.1)
3474:boundary
3099:are the
1844:) via a
4466:Bibcode
4282:sin(10)
3799:is the
2452:versus
2118:versus
1756:versus
212:is the
195:complex
75:of the
35:at the
4591:
4566:
4484:
4440:
4394:: the
4297:series
4028:|
3470:circle
3453:|
3179:= cos(
2876:arctan
1534:where
857:is an
457:
401:
393:
193:is a
186:where
23:, the
4660:Radii
4482:S2CID
4402:Notes
3784:with
3468:is a
3372:) = 1
3193:then
3113:other
27:of a
4589:ISBN
4564:ISBN
4438:ISBN
3994:<
3368:sin(
3364:) +
3360:cos(
3332:sin(
3328:) +
3324:cos(
3314:and
3187:sin(
3183:) +
3174:and
1133:<
1085:<
494:| =
328:>
280:<
199:disk
67:and
33:disk
4508:doi
4474:doi
4462:240
4278:= 1
4059:sin
3937:log
3673:of
3656:= 1
3582:= 0
3500:= 0
3277:sin
3256:cos
2536:cos
1631:lim
1551:lim
1219:lim
1143:lim
966:lim
886:lim
366:sup
19:In
4641::
4587:,
4562:,
4537:29
4535:.
4531:.
4504:50
4502:,
4480:,
4472:,
4460:,
3780:)/
3684:.
3458:−
3427:.
3171:iy
3169:+
3165:=
2853:.
2599:.
1443:2.
1088:1.
861:.
509:.
4510::
4476::
4468::
4391:n
4387:a
4383:s
4366:.
4359:s
4355:n
4349:n
4345:a
4332:1
4329:=
4326:n
4276:x
4243:x
4226:x
4209:!
4206:5
4200:5
4196:x
4190:+
4184:!
4181:3
4175:3
4171:x
4162:x
4159:=
4154:1
4151:+
4148:n
4145:2
4141:x
4134:!
4131:)
4128:1
4125:+
4122:n
4119:2
4116:(
4109:n
4105:)
4101:1
4095:(
4082:0
4079:=
4076:n
4068:=
4065:x
4031:z
4007:,
4002:n
3998:2
3991:i
3983:1
3977:n
3973:2
3964:1
3961:+
3955:)
3952:i
3949:(
3941:2
3930:=
3927:n
3916:n
3911:n
3907:2
3899:1
3893:n
3889:)
3885:1
3879:(
3873:=
3868:i
3864:a
3853:i
3849:z
3843:i
3839:a
3828:1
3825:=
3822:i
3797:)
3795:z
3793:(
3791:h
3786:g
3782:z
3778:z
3776:(
3774:g
3770:z
3768:(
3766:h
3761:h
3741:n
3737:z
3729:2
3725:n
3721:1
3709:1
3706:=
3703:n
3682:)
3680:z
3678:(
3676:g
3667:)
3665:z
3663:(
3661:f
3654:z
3636:,
3631:n
3627:z
3621:n
3618:1
3606:1
3603:=
3600:n
3580:z
3575:)
3573:z
3569:z
3567:(
3565:g
3544:,
3539:n
3535:z
3524:0
3521:=
3518:n
3498:z
3493:)
3491:z
3487:z
3485:(
3483:f
3465:r
3460:a
3456:z
3448:z
3442:r
3437:a
3425:π
3421:π
3417:i
3414:π
3408:.
3406:i
3403:π
3399:z
3392:π
3388:y
3384:y
3380:y
3376:y
3370:y
3366:i
3362:y
3356:z
3352:x
3348:x
3344:e
3340:e
3336:)
3334:y
3330:i
3326:y
3320:y
3316:y
3312:x
3295:,
3292:)
3289:)
3286:y
3283:(
3274:i
3271:+
3268:)
3265:y
3262:(
3253:(
3248:x
3244:e
3240:=
3235:y
3232:i
3228:e
3222:x
3218:e
3214:=
3209:z
3205:e
3191:)
3189:y
3185:i
3181:y
3177:e
3167:x
3163:z
3145:0
3142:=
3139:1
3131:z
3127:e
3105:z
3096:n
3092:B
3073:n
3069:z
3062:!
3059:n
3053:n
3049:B
3036:0
3033:=
3030:n
3022:=
3016:1
3008:z
3004:e
2999:z
2963:.
2957:+
2952:7
2947:7
2943:z
2932:5
2927:5
2923:z
2917:+
2912:3
2907:3
2903:z
2894:z
2891:=
2888:)
2885:z
2882:(
2844:i
2840:z
2838:(
2836:f
2819:.
2814:n
2811:2
2807:z
2801:n
2797:)
2793:1
2787:(
2777:0
2774:=
2771:n
2741:2
2737:z
2733:+
2730:1
2702:2
2698:z
2694:+
2691:1
2687:1
2682:=
2679:)
2676:z
2673:(
2670:f
2648:.
2631:2
2627:z
2623:+
2620:1
2616:1
2593:a
2585:f
2581:a
2577:a
2573:f
2554:.
2516:0
2513:=
2508:2
2504:n
2499:/
2495:1
2473:2
2469:n
2464:/
2460:1
2439:)
2430:n
2426:b
2420:n
2416:c
2409:1
2406:+
2403:n
2399:c
2393:+
2386:n
2382:c
2375:n
2371:b
2365:1
2359:n
2355:c
2347:(
2341:2
2338:1
2316:r
2312:/
2308:)
2305:1
2302:+
2299:p
2296:(
2250:p
2230:r
2226:/
2222:1
2202:0
2199:=
2196:n
2192:/
2188:1
2168:0
2165:=
2162:n
2158:/
2154:1
2134:n
2130:/
2126:1
2104:n
2100:b
2079:.
2073:,
2070:5
2067:,
2064:4
2061:,
2058:3
2055:=
2052:n
2043:2
2038:1
2032:n
2028:c
2019:2
2013:n
2009:c
2003:n
1999:c
1991:2
1986:n
1982:c
1973:1
1967:n
1963:c
1957:1
1954:+
1951:n
1947:c
1940:=
1935:2
1930:n
1926:b
1914:.
1898:r
1894:/
1890:1
1870:0
1867:=
1864:n
1860:/
1856:1
1829:=
1826:n
1806:0
1803:=
1800:n
1796:/
1792:1
1772:n
1768:/
1764:1
1742:1
1736:n
1732:c
1727:/
1721:n
1717:c
1696:r
1673:1
1667:n
1663:c
1658:/
1652:n
1648:c
1635:n
1627:=
1624:r
1620:/
1616:1
1593:1
1587:n
1583:c
1578:/
1572:n
1568:c
1555:n
1536:r
1522:r
1518:/
1514:1
1493:n
1471:n
1467:c
1439:/
1435:1
1432:=
1429:r
1409:2
1405:/
1401:1
1395:=
1364:.
1355:2
1352:+
1349:1
1344:)
1339:3
1331:+
1328:1
1325:(
1316:=
1313:)
1307:(
1304:f
1272:.
1268:|
1261:1
1258:+
1255:n
1251:c
1245:n
1241:c
1235:|
1223:n
1215:=
1205:|
1199:n
1195:c
1190:|
1183:|
1177:1
1174:+
1171:n
1167:c
1162:|
1147:n
1138:1
1129:|
1125:a
1119:z
1115:|
1078:|
1072:n
1068:)
1064:a
1058:z
1055:(
1050:n
1046:c
1041:|
1034:|
1028:1
1025:+
1022:n
1018:)
1014:a
1008:z
1005:(
1000:1
997:+
994:n
990:c
985:|
970:n
939:.
935:|
928:1
925:+
922:n
918:c
912:n
908:c
902:|
890:n
882:=
879:r
855:f
851:r
824:n
818:|
812:n
808:c
803:|
788:n
779:1
774:=
771:r
758:a
754:z
750:C
746:C
724:|
720:a
714:z
710:|
705:)
699:n
693:|
687:n
683:c
678:|
671:(
659:n
651:=
645:n
639:|
633:n
629:)
625:a
619:z
616:(
611:n
607:c
602:|
587:n
579:=
576:C
534:n
530:c
507:z
500:z
496:r
492:a
488:z
470:}
452:n
448:)
444:a
438:z
435:(
430:n
426:c
415:0
412:=
409:n
397:|
389:|
385:a
379:z
375:|
370:{
363:=
360:r
334:.
331:r
324:|
320:a
314:z
310:|
283:r
276:|
272:a
266:z
262:|
228:r
220:z
214:n
209:n
205:c
191:a
171:,
166:n
162:)
158:a
152:z
149:(
144:n
140:c
129:0
126:=
123:n
115:=
112:)
109:z
106:(
103:f
90:f
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