1794:
2892:
2098:
6648:
2887:{\displaystyle {\begin{aligned}e^{ix}&=1+ix+{\frac {(ix)^{2}}{2!}}+{\frac {(ix)^{3}}{3!}}+{\frac {(ix)^{4}}{4!}}+{\frac {(ix)^{5}}{5!}}+{\frac {(ix)^{6}}{6!}}+{\frac {(ix)^{7}}{7!}}+{\frac {(ix)^{8}}{8!}}+\cdots \\&=1+ix-{\frac {x^{2}}{2!}}-{\frac {ix^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {ix^{5}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {ix^{7}}{7!}}+{\frac {x^{8}}{8!}}+\cdots \\&=\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}-\cdots \right)+i\left(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \right)\\&=\cos x+i\sin x,\end{aligned}}}
6307:
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4454:
5820:
8303:
5287:
3432:
52:
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538:
2088:
6302:{\displaystyle {\begin{aligned}\cos nx&=\operatorname {Re} \left(e^{inx}\right)\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot e^{ix}\right)\\&=\operatorname {Re} {\Big (}e^{i(n-1)x}\cdot {\big (}\underbrace {e^{ix}+e^{-ix}} _{2\cos x}-e^{-ix}{\big )}{\Big )}\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot 2\cos x-e^{i(n-2)x}\right)\\&=\cos\cdot -\cos.\end{aligned}}}
5804:{\displaystyle {\begin{aligned}\cos x\cos y&={\frac {e^{ix}+e^{-ix}}{2}}\cdot {\frac {e^{iy}+e^{-iy}}{2}}\\&={\frac {1}{2}}\cdot {\frac {e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2}}\\&={\frac {1}{2}}{\bigg (}{\frac {e^{i(x+y)}+e^{-i(x+y)}}{2}}+{\frac {e^{i(x-y)}+e^{-i(x-y)}}{2}}{\bigg )}\\&={\frac {1}{2}}\left(\cos(x+y)+\cos(x-y)\right).\end{aligned}}}
4686:
1825:
3814:
5089:
4477:
5273:
3194:
1691:
1535:
1247:
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5277:
Complex exponentials can simplify trigonometry, because they are mathematically easier to manipulate than their sine and cosine components. One technique is simply to convert sines and cosines into equivalent expressions in terms of exponentials sometimes called
4878:
4290:
2083:{\displaystyle {\begin{aligned}i^{0}&=1,&i^{1}&=i,&i^{2}&=-1,&i^{3}&=-i,\\i^{4}&=1,&i^{5}&=i,&i^{6}&=-1,&i^{7}&=-i\\&\vdots &&\vdots &&\vdots &&\vdots \end{aligned}}}
7636:
765:
641:
turned his attention to the exponential function and derived the equation named after him by comparing the series expansions of the exponential and trigonometric expressions. The formula was first published in 1748 in his foundational work
7059:
5096:
3043:
3417:
7895:
Bernoulli, Johann (1702). "Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul" [Solution of a problem in integral calculus with some notes relating to this calculation].
933:
and using the complex algebraic operations. In particular, we may use any of the three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the
1547:
526:
Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of
7765:
1364:
1412:
7492:. In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms a measure of angle into circular arc length (here, the modulus is the radius (
1096:
4461:
Euler's formula, the definitions of the trigonometric functions and the standard identities for exponentials are sufficient to easily derive most trigonometric identities. It provides a powerful connection between
2103:
859:
4693:
4681:{\displaystyle {\begin{aligned}\cos x&=\operatorname {Re} \left(e^{ix}\right)={\frac {e^{ix}+e^{-ix}}{2}},\\\sin x&=\operatorname {Im} \left(e^{ix}\right)={\frac {e^{ix}-e^{-ix}}{2i}}.\end{aligned}}}
3018:
4172:
1396:
This proof shows that the quotient of the trigonometric and exponential expressions is the constant function one, so they must be equal (the exponential function is never zero, so this is permitted).
1784:
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322:
3809:{\displaystyle {\begin{aligned}z&=x+iy=|z|(\cos \varphi +i\sin \varphi )=re^{i\varphi },\\{\bar {z}}&=x-iy=|z|(\cos \varphi -i\sin \varphi )=re^{-i\varphi },\end{aligned}}}
4159:
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5084:{\displaystyle {\begin{aligned}\cos iy&={\frac {e^{-y}+e^{y}}{2}}=\cosh y,\\\sin iy&={\frac {e^{-y}-e^{y}}{2i}}={\frac {e^{y}-e^{-y}}{2}}i=i\sinh y.\end{aligned}}}
4100:
631:
585:
459:
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6638:
995:
6676:
is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the
1081:
559:
1287:
605:
6647:
214:
772:
6969:. This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging the
2950:
3028:; they will be determined in the course of the proof. From any of the definitions of the exponential function it can be shown that the derivative of
7699:
5268:{\displaystyle {\begin{aligned}\cosh ix&={\frac {e^{ix}+e^{-ix}}{2}}=\cos x,\\\sinh ix&={\frac {e^{ix}-e^{-ix}}{2}}=i\sin x.\end{aligned}}}
3558:
3189:{\displaystyle ie^{ix}=\left(\cos \theta +i\sin \theta \right){\frac {dr}{dx}}+r\left(-\sin \theta +i\cos \theta \right){\frac {d\theta }{dx}}.}
1729:
20:
912:
7873:
4302:
1686:{\displaystyle f'(\theta )=e^{-i\theta }\left(i\cos \theta -\sin \theta \right)-ie^{-i\theta }\left(\cos \theta +i\sin \theta \right)=0}
8288:
7101:
541:
Visualization of Euler's formula as a helix in three-dimensional space. The helix is formed by plotting points for various values of
4375:
8153:
1530:{\displaystyle f(\theta )={\frac {\cos \theta +i\sin \theta }{e^{i\theta }}}=e^{-i\theta }\left(\cos \theta +i\sin \theta \right)}
6695:, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see
1242:{\displaystyle e^{z}=1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}
207:
8198:
7496:) of the circle). According to Cotes, the product of the modulus and the measure (logarithm) of the ratio, when multiplied by
8329:
8213:
8102:
8083:
7854:
7444:
6737:
464:
267:
8238:
7091:
892:
8126:
4060:
of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation):
8248:
7985:
7847:
6381:
4851:{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x,\\e^{-ix}&=\cos(-x)+i\sin(-x)=\cos x-i\sin x\end{aligned}}}
4105:
3565:. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number
644:
7869:
1008:
8339:
7924:
7096:
868:
Bernoulli's correspondence with Euler (who also knew the above equation) shows that
Bernoulli did not fully understand
200:
32:
7995:
7968:
7825:
7158:
7131:
6407:
150:
6389:
8258:
4285:{\displaystyle z=\left|z\right|e^{i\varphi }=e^{\ln \left|z\right|}e^{i\varphi }=e^{\ln \left|z\right|+i\varphi }}
4063:
865:
by relating natural logarithms to imaginary (complex) numbers. Bernoulli, however, did not evaluate the integral.
561:
and is determined by both the cosine and sine components of the formula. One curve represents the real component (
8283:
8228:
896:
6385:
8045:
8193:
7944:), How to prove Euler's formula: $ e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$ ?, URL (version: 2018-06-25):
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142:
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8203:
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8173:
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7398:
7285:
7194:
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133:
6552:
6497:
587:) of the formula, while another curve, rotated 90 degrees around the z-axis (due to multiplication by
8188:
6427:
4861:
These formulas can even serve as the definition of the trigonometric functions for complex arguments
2895:
329:
42:
7631:{\displaystyle {\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta }
760:{\displaystyle {\frac {1}{1+x^{2}}}={\frac {1}{2}}\left({\frac {1}{1-ix}}+{\frac {1}{1+ix}}\right).}
8218:
8178:
8012:
6370:
4438:
188:
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7641:
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7331:
7240:
6707:
of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.
6526:
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610:
564:
437:
8349:
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8139:
6614:
6374:
3869:
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998:
434:
presented a geometrical argument that can be interpreted (after correcting a misplaced factor of
387:
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist
349:
246:
958:
8278:
7054:{\displaystyle {\begin{aligned}e^{i\tau }&=\cos \tau +i\sin \tau \\&=1+0\end{aligned}}}
6711:
6688:
4442:
2913:
7958:
7815:
5817:
of a complex expression and perform the manipulations on the complex expression. For example:
8344:
7791:
7665:
should be on the right side of the equation, not the left side. If the change of scaling by
6667:
6337:
4463:
4362:
1260:
938:
544:
7842:
7121:
6661:
8112:
3508:
2917:
253:
71:
7395:, which is the "sinus of the complement to the quadrant" or "cosinus". The ratio between
8:
8233:
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6641:
6579:
6461:
3469:
419:
92:
7148:
8253:
8072:
3493:
590:
108:
7941:
6699:), and these are more conveniently expressed as the sum of exponential functions with
3412:{\displaystyle e^{i\theta }=1(\cos \theta +i\sin \theta )=\cos \theta +i\sin \theta .}
8334:
8098:
8079:
7991:
7964:
7920:
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7154:
7127:
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6469:
4358:
3909:
2929:
869:
862:
361:
112:
87:
66:
5282:. After the manipulations, the simplified result is still real-valued. For example:
7876:, page 214, section 138 (translation by Ian Bruce, pdf link from 17 century maths).
6700:
6696:
6583:
3561:. Euler's formula provides a means of conversion between cartesian coordinates and
1373:
1002:
935:
651:
242:
4690:
The two equations above can be derived by adding or subtracting Euler's formulas:
391:
called the equation "our jewel" and "the most remarkable formula in mathematics".
8108:
6704:
3492:
that a line connecting the origin with a point on the unit circle makes with the
2092:
Using now the power-series definition from above, we see that for real values of
388:
4051:
2928:
Another proof is based on the fact that all complex numbers can be expressed in
872:. Euler also suggested that complex logarithms can have infinitely many values.
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638:
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250:
231:
171:
19:
This article is about Euler's formula in complex analysis. For other uses, see
8323:
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8037:
7760:{\displaystyle \cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }}
6677:
3554:
3504:
3477:
1815:
1793:
880:
876:
3542:). In fact, the same proof shows that Euler's formula is even valid for all
1359:{\displaystyle e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.}
915:). Several of these methods may be directly extended to give definitions of
8268:
6959:
6813:
that evaluate to units illustrate rotation around the complex unit circle:
6810:
4471:
4467:
4056:
Now, taking this derived formula, we can use Euler's formula to define the
1542:
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431:
257:
235:
166:
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1252:
8095:
Euler's
Pioneering Equation: The Most Beautiful Theorem in Mathematics
5814:
4470:, and provides an interpretation of the sine and cosine functions as
4057:
359:
respectively. This complex exponential function is sometimes denoted
102:
7266:(Thus if any arc of a quadrant of a circle, described by the radius
6359:
5813:
Another technique is to represent sines and cosines in terms of the
3431:
51:
7780:(338) : 5-45; see especially page 32. Available on-line at:
6798:
6465:
6421:
2933:
8041:
854:{\displaystyle \int {\frac {dx}{1+ax}}={\frac {1}{a}}\ln(1+ax)+C,}
537:
8131:
6424:, Euler's formula states that the imaginary exponential function
893:
Exponentiation § Complex exponents with a positive real base
238:
3013:{\displaystyle e^{ix}=r\left(\cos \theta +i\sin \theta \right).}
7797:
6970:
6794:
6719:
3925:
3497:
352:
7870:
Chapter 8: On transcending quantities arising from the circle
7282:
as modulus, the arc will be the measure of the ratio between
4052:
Use of the formula to define the logarithm of complex numbers
3929:
3888:
3489:
1376:, so there is no question about what the power with exponent
3462:
This formula can be interpreted as saying that the function
3221:
and equating real and imaginary parts in this formula gives
6640:. These observations may be combined and summarized in the
6336:
a parameter in equation above yields recursive formula for
356:
8074:
Dr. Euler's
Fabulous Formula: Cures Many Mathematical Ills
6651:
Euler's formula and identity combined in diagrammatic form
4457:
Relationship between sine, cosine and exponential function
2894:
where in the last step we recognize the two terms are the
245:
that establishes the fundamental relationship between the
7774:
21:
List of things named after
Leonhard Euler § Formulas
7790:… (Cambridge, England: 1722), chapter: "Logometria",
7638:. This equation has a misplaced factor: the factor of
4357:
and in fact, this can be used as the definition for the
3426:
1779:{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta .}
886:
7485:{\displaystyle \cos \theta +{\sqrt {-1}}\sin \theta \ }
911:
may be defined in a few different equivalent ways (see
1797:
A plot of the first few terms of the Taylor series of
7702:
7671:
7644:
7541:
7523:, equals the length of the circular arc subtended by
7502:
7447:
7401:
7375:
as the other side. The perpendicular from the point
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270:
7371:
having the positive x-axis as one side and a radius
7942:
https://math.stackexchange.com/users/2720/user02138
7898:
Mémoires de l'Académie Royale des
Sciences de Paris
4350:{\displaystyle \ln z=\ln \left|z\right|+i\varphi ,}
950:
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3557:can be represented by a complex number written in
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316:
7175:"Nam si quadrantis circuli quilibet arcus, radio
6311:This formula is used for recursive generation of
6093:
5965:
5714:
5574:
4437:, together with Euler's formula, implies several
8321:
4299:. Taking the logarithm of both sides shows that
3457:
1305:
897:Exponential function § On the complex plane
7960:A Modern Introduction to Differential Equations
7696:and exponentiating both sides, the result is:
6797:in quaternions. The set of all versors forms a
4448:
3978:, but the first equation needs adjustment when
16:Complex exponential in terms of sine and cosine
7813:
7359:(at the origin of the (x,y) plane) and radius
4426:{\displaystyle \left(e^{a}\right)^{k}=e^{ak},}
4361:. The logarithm of a complex number is thus a
3040:. Therefore, differentiating both sides gives
8147:
7527:, which for any angle measured in radians is
7383: ; the line between the circle's center
7126:. World Scientific Publishing Co. p. 7.
6349:
6086:
6003:
1818:, as well as basic facts about the powers of
1005:for which the derivative equals the function
913:Characterizations of the exponential function
875:The view of complex numbers as points in the
208:
7874:Introduction to the Analysis of the Infinite
7692:is made, then, after dividing both sides by
7355:.) That is, consider a circle having center
7274:and sinus of the complement to the quadrant
7123:A Course in Complex Analysis in One Variable
6646:
1388:Various proofs of the formula are possible.
861:the above equation tells us something about
7809:
7807:
7379:on the circle to the x-axis is the "sinus"
6388:. Unsourced material may be challenged and
4433:which can be seen to hold for all integers
3993:, not both zero, the angles of the vectors
8154:
8140:
8030:
2923:
1085:
215:
201:
7894:
7119:
7102:List of things named after Leonhard Euler
7061:An interpretation of the simplified form
6597:
6558:
6531:
6503:
6480:
6408:Learn how and when to remove this message
4021:radians, but have the identical value of
1814:Here is a proof of Euler's formula using
7804:
6734:a real number, Euler's formula applies:
6703:exponents, using Euler's formula. Also,
4452:
3430:
1792:
1391:
536:
7956:
7914:
7150:The Feynman Lectures on Physics, vol. I
7146:
6786:{\displaystyle \exp xr=\cos x+r\sin x,}
4858:and solving for either cosine or sine.
3539:
2914:The rearrangement of terms is justified
607:), represents the imaginary component (
519:{\displaystyle ix=\ln(\cos x+i\sin x).}
256:. Euler's formula states that, for any
8322:
8092:
7983:
7977:
7908:
7191:pro Modulo, arcus erit rationis inter
7068:is that rotating by a full turn is an
6804:
3484:ranges through the real numbers. Here
879:was described about 50 years later by
404:, Euler's formula may be rewritten as
372:sine"). The formula is still valid if
317:{\displaystyle e^{ix}=\cos x+i\sin x,}
8135:
8069:
8036:
7946:https://math.stackexchange.com/q/8612
6655:
3427:Applications in complex number theory
1788:
887:Definitions of complex exponentiation
7990:. Wellesley-Cambridge. p. 389.
7786:Roger Cotes with Robert Smith, ed.,
7391:at the foot of the perpendicular is
6386:adding citations to reliable sources
6353:
3020:No assumptions are being made about
8097:. Oxford: Oxford University Press.
7957:Ricardo, Henry J. (23 March 2016).
7848:Mathematical Association of America
7183:sinumque complementi ad quadrantem
4372:Finally, the other exponential law
4161:both valid for any complex numbers
4154:{\displaystyle e^{a}e^{b}=e^{a+b},}
3503:The original proof is based on the
3496:, measured counterclockwise and in
1276:
1255:, it is possible to show that this
645:Introductio in analysin infinitorum
430:In 1714, the English mathematician
13:
8214:Euler's continued fraction formula
8161:
8063:
7097:History of Lorentz transformations
6662:Complex number § Applications
1315:
1209:
1041:{\displaystyle {\frac {df}{dz}}=f}
14:
8361:
8239:Euler's pump and turbine equation
8120:
8048:from the original on 28 June 2019
7963:. Elsevier Science. p. 428.
7772:Roger Cotes (1714) "Logometria,"
7430:{\displaystyle EX+XC{\sqrt {-1}}}
7317:{\displaystyle EX+XC{\sqrt {-1}}}
7226:{\displaystyle EX+XC{\sqrt {-1}}}
7153:. Addison-Wesley. p. 22-10.
7092:Integration using Euler's formula
6604:{\displaystyle \tau \mathbb {Z} }
3453:illustrated in the complex plane.
8302:
8301:
8259:Euler equations (fluid dynamics)
8249:Euler's sum of powers conjecture
6571:{\displaystyle \mathbb {S} ^{1}}
6516:{\displaystyle \mathbb {S} ^{1}}
6358:
951:Differential equation definition
50:
8005:
7950:
7934:
6453:{\displaystyle t\mapsto e^{it}}
3985:. This is because for any real
3421:
1541:. Differentiating gives by the
8199:Euler–Poisson–Darboux equation
8078:. Princeton University Press.
7888:
7879:
7859:
7834:
7622:
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7167:
7140:
7113:
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3916:, i.e., the angle between the
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510:
483:
1:
7885:Conway & Guy, pp. 254–255
7120:Moskowitz, Martin A. (2002).
7107:
4474:of the exponential function:
3924:measured counterclockwise in
3579:, and its complex conjugate,
3458:Interpretation of the formula
330:base of the natural logarithm
151:Lindemann–Weierstrass theorem
8330:Theorems in complex analysis
8229:Euler's four-square identity
7840:Sandifer, C. Edward (2007),
7798:https://nrich.maths.org/1384
7685:{\displaystyle {\sqrt {-1}}}
7658:{\displaystyle {\sqrt {-1}}}
7516:{\displaystyle {\sqrt {-1}}}
7348:{\displaystyle {\sqrt {-1}}}
7257:{\displaystyle {\sqrt {-1}}}
7147:Feynman, Richard P. (1977).
6793:and the element is called a
6538:{\displaystyle \mathbb {R} }
6487:{\displaystyle \mathbb {R} }
4449:Relationship to trigonometry
4169:. Therefore, one can write:
4095:{\displaystyle a=e^{\ln a},}
3520:is a complex number) and of
626:{\displaystyle \sin \theta }
580:{\displaystyle \cos \theta }
454:{\displaystyle {\sqrt {-1}}}
7:
8284:Euler–Bernoulli beam theory
7817:Mathematics and Its History
7767:, which is Euler's formula.
7075:
6633:{\displaystyle \tau =2\pi }
384:in this more general case.
10:
8366:
7278: ; taking the radius
6659:
6350:Topological interpretation
6342:
3472:, i.e., it traces out the
3323:. This proves the formula
990:{\displaystyle f(z)=e^{z}}
890:
425:
18:
8340:E (mathematical constant)
8297:
8194:Euler–Mascheroni constant
8169:
7179:descriptus, sinun habeat
6523:. In fact, this exhibits
1383:
955:The exponential function
901:The exponential function
8244:Euler's rotation theorem
7984:Strang, Gilbert (1991).
6345:Phasor § Arithmetic
4439:trigonometric identities
8204:Euler–Rodrigues formula
8184:Euler–Maclaurin formula
8174:Euler–Lagrange equation
8070:Nahin, Paul J. (2006).
7919:. Pearson. p. 20.
7814:John Stillwell (2002).
7187: ; sumendo radium
6973:from the general case:
4865:. For example, letting
2924:Using polar coordinates
2916:because each series is
1816:power-series expansions
1086:Power series definition
1076:{\displaystyle f(0)=1.}
999:differentiable function
925:simply by substituting
554:{\displaystyle \theta }
350:trigonometric functions
247:trigonometric functions
8279:Euler number (physics)
8209:Euler–Tricomi equation
8093:Wilson, Robin (2018).
7761:
7686:
7659:
7632:
7517:
7486:
7431:
7349:
7318:
7258:
7227:
7055:
6787:
6712:four-dimensional space
6689:electrical engineering
6668:differential equations
6652:
6634:
6605:
6572:
6539:
6517:
6488:
6454:
6318:for integer values of
6303:
5805:
5269:
5085:
4852:
4682:
4458:
4427:
4351:
4286:
4155:
4096:
3841:is the imaginary part,
3810:
3454:
3413:
3190:
3014:
2888:
2084:
1811:
1780:
1687:
1531:
1399:Consider the function
1360:
1243:
1213:
1077:
1042:
991:
947:to the complex plane.
921:for complex values of
855:
761:
634:
627:
601:
581:
555:
520:
455:
318:
40:mathematical constant
8219:Euler's critical load
8189:Euler–Maruyama method
8002:Second proof on page.
7917:Mathematical Analysis
7915:Apostol, Tom (1974).
7843:Euler's Greatest Hits
7762:
7687:
7660:
7633:
7518:
7487:
7432:
7363:. Consider an angle
7350:
7319:
7259:
7228:
7056:
6788:
6650:
6635:
6606:
6573:
6540:
6518:
6489:
6455:
6338:Chebyshev polynomials
6304:
5806:
5270:
5086:
4853:
4683:
4456:
4428:
4363:multi-valued function
4352:
4287:
4156:
4097:
3811:
3559:cartesian coordinates
3434:
3414:
3285:. The initial values
3191:
3015:
2918:absolutely convergent
2889:
2085:
1796:
1781:
1704:is a constant. Since
1688:
1532:
1392:Using differentiation
1361:
1261:radius of convergence
1244:
1193:
1078:
1043:
992:
939:analytic continuation
891:Further information:
856:
762:
628:
602:
582:
556:
540:
521:
456:
380:, and is also called
319:
189:Schanuel's conjecture
8179:Euler–Lotka equation
7700:
7669:
7642:
7539:
7500:
7445:
7399:
7332:
7286:
7241:
7195:
6977:
6938:The special case at
6738:
6730:on this sphere, and
6680:of the operation of
6615:
6590:
6553:
6527:
6498:
6476:
6428:
6382:improve this section
5821:
5288:
5097:
4879:
4694:
4478:
4376:
4303:
4173:
4106:
4064:
3920:axis and the vector
3599:
3595:, can be written as
3509:exponential function
3327:
3044:
2951:
2099:
1826:
1730:
1548:
1413:
1288:
1097:
1052:
1009:
959:
773:
658:
611:
591:
565:
545:
465:
438:
418:, which is known as
268:
254:exponential function
72:Exponential function
33:a series of articles
8127:Elements of Algebra
8042:"The Tau Manifesto"
8013:"Complex Sinusoids"
7788:Harmonia mensurarum
7367:with its vertex at
6805:Other special cases
6642:commutative diagram
6494:to the unit circle
6472:from the real line
6420:In the language of
6340:of the first kind.
4443:de Moivre's formula
3939:. Many texts write
3928:, which is defined
3470:unit complex number
3267:is a constant, and
907:for real values of
143:representations of
8040:(14 March 2019) .
8017:ccrma.stanford.edu
7757:
7682:
7655:
7628:
7513:
7482:
7427:
7345:
7314:
7254:
7223:
7051:
7049:
6783:
6656:Other applications
6653:
6630:
6601:
6568:
6535:
6513:
6484:
6470:topological groups
6450:
6299:
6297:
6064:
6048:
5801:
5799:
5265:
5263:
5081:
5079:
4848:
4846:
4678:
4676:
4459:
4423:
4347:
4282:
4151:
4092:
3806:
3804:
3507:expansions of the
3494:positive real axis
3455:
3409:
3281:for some constant
3186:
3010:
2884:
2882:
2080:
2078:
1812:
1789:Using power series
1776:
1683:
1527:
1356:
1319:
1239:
1073:
1038:
987:
870:complex logarithms
863:complex logarithms
851:
757:
635:
623:
597:
577:
551:
516:
451:
314:
8317:
8316:
8104:978-0-19-879492-9
8085:978-0-691-11822-2
7855:978-0-88385-563-8
7750:
7723:
7680:
7653:
7608:
7590:
7550:
7511:
7481:
7468:
7425:
7343:
7312:
7252:
7237:mensura ducta in
7221:
7070:identity function
6936:
6935:
6693:signal processing
6418:
6417:
6410:
6010:
6008:
5737:
5710:
5642:
5570:
5550:
5423:
5403:
5359:
5280:complex sinusoids
5241:
5162:
5054:
5016:
4938:
4669:
4570:
4369:is multi-valued.
4359:complex logarithm
3829:is the real part,
3709:
3563:polar coordinates
3534:for real numbers
3181:
3118:
2930:polar coordinates
2830:
2805:
2780:
2730:
2705:
2680:
2655:
2606:
2581:
2551:
2526:
2496:
2471:
2441:
2388:
2352:
2316:
2280:
2244:
2208:
2172:
1472:
1374:positive integers
1372:is restricted to
1340:
1304:
1234:
1182:
1157:
1132:
1030:
816:
803:
747:
723:
697:
684:
600:{\displaystyle i}
449:
225:
224:
88:compound interest
67:Natural logarithm
8357:
8305:
8304:
8234:Euler's identity
8156:
8149:
8142:
8133:
8132:
8116:
8089:
8077:
8058:
8057:
8055:
8053:
8034:
8028:
8027:
8025:
8023:
8009:
8003:
8001:
7981:
7975:
7974:
7954:
7948:
7938:
7932:
7930:
7912:
7906:
7905:
7892:
7886:
7883:
7877:
7863:
7857:
7838:
7832:
7831:
7811:
7802:
7766:
7764:
7763:
7758:
7756:
7755:
7751:
7743:
7724:
7716:
7691:
7689:
7688:
7683:
7681:
7673:
7664:
7662:
7661:
7656:
7654:
7646:
7637:
7635:
7634:
7629:
7609:
7606:
7605:
7601:
7591:
7583:
7551:
7543:
7522:
7520:
7519:
7514:
7512:
7504:
7491:
7489:
7488:
7483:
7479:
7469:
7461:
7436:
7434:
7433:
7428:
7426:
7418:
7354:
7352:
7351:
7346:
7344:
7336:
7323:
7321:
7320:
7315:
7313:
7305:
7263:
7261:
7260:
7255:
7253:
7245:
7232:
7230:
7229:
7224:
7222:
7214:
7171:
7165:
7164:
7144:
7138:
7137:
7117:
7087:Euler's identity
7067:
7060:
7058:
7057:
7052:
7050:
7031:
6996:
6995:
6968:
6957:
6947:
6932:
6924:
6919:
6917:
6916:
6913:
6910:
6897:
6892:
6879:
6872:
6867:
6865:
6864:
6861:
6858:
6845:
6840:
6830:
6823:
6816:
6815:
6801:in the 4-space.
6792:
6790:
6789:
6784:
6733:
6729:
6726:. For any point
6697:Fourier analysis
6675:
6639:
6637:
6636:
6631:
6610:
6608:
6607:
6602:
6600:
6580:Euler's identity
6577:
6575:
6574:
6569:
6567:
6566:
6561:
6544:
6542:
6541:
6536:
6534:
6522:
6520:
6519:
6514:
6512:
6511:
6506:
6493:
6491:
6490:
6485:
6483:
6459:
6457:
6456:
6451:
6449:
6448:
6413:
6406:
6402:
6399:
6393:
6362:
6354:
6335:
6325:
6321:
6317:
6308:
6306:
6305:
6300:
6298:
6201:
6197:
6193:
6192:
6191:
6146:
6145:
6101:
6097:
6096:
6090:
6089:
6083:
6082:
6063:
6049:
6044:
6043:
6042:
6024:
6023:
6007:
6006:
5997:
5996:
5969:
5968:
5950:
5946:
5942:
5941:
5940:
5925:
5924:
5880:
5876:
5872:
5871:
5810:
5808:
5807:
5802:
5800:
5793:
5789:
5738:
5730:
5722:
5718:
5717:
5711:
5706:
5705:
5704:
5674:
5673:
5648:
5643:
5638:
5637:
5636:
5606:
5605:
5580:
5578:
5577:
5571:
5563:
5555:
5551:
5546:
5545:
5544:
5514:
5513:
5483:
5482:
5455:
5454:
5429:
5424:
5416:
5408:
5404:
5399:
5398:
5397:
5379:
5378:
5365:
5360:
5355:
5354:
5353:
5335:
5334:
5321:
5274:
5272:
5271:
5266:
5264:
5242:
5237:
5236:
5235:
5217:
5216:
5203:
5163:
5158:
5157:
5156:
5138:
5137:
5124:
5090:
5088:
5087:
5082:
5080:
5055:
5050:
5049:
5048:
5033:
5032:
5022:
5017:
5015:
5007:
5006:
5005:
4993:
4992:
4979:
4939:
4934:
4933:
4932:
4920:
4919:
4906:
4874:
4864:
4857:
4855:
4854:
4849:
4847:
4767:
4766:
4713:
4712:
4687:
4685:
4684:
4679:
4677:
4670:
4668:
4660:
4659:
4658:
4640:
4639:
4626:
4621:
4617:
4616:
4571:
4566:
4565:
4564:
4546:
4545:
4532:
4527:
4523:
4522:
4436:
4432:
4430:
4429:
4424:
4419:
4418:
4403:
4402:
4397:
4393:
4392:
4368:
4356:
4354:
4353:
4348:
4334:
4298:
4291:
4289:
4288:
4283:
4281:
4280:
4270:
4245:
4244:
4232:
4231:
4230:
4205:
4204:
4192:
4168:
4164:
4160:
4158:
4157:
4152:
4147:
4146:
4128:
4127:
4118:
4117:
4101:
4099:
4098:
4093:
4088:
4087:
4047:
4046:
4044:
4043:
4038:
4035:
4020:
4016:
4004:
3992:
3988:
3984:
3977:
3962:
3960:
3958:
3957:
3954:
3951:
3938:
3915:
3907:
3900:
3875:
3867:
3866:
3865:
3853:
3840:
3828:
3815:
3813:
3812:
3807:
3805:
3798:
3797:
3746:
3738:
3711:
3710:
3702:
3692:
3691:
3643:
3635:
3594:
3585:
3578:
3549:
3537:
3533:
3526:
3519:
3515:
3487:
3483:
3467:
3452:
3435:Euler's formula
3418:
3416:
3415:
3410:
3342:
3341:
3322:
3312:
3305:
3298:
3291:
3284:
3280:
3270:
3266:
3262:
3259:
3257:
3256:
3253:
3250:
3241:
3238:
3236:
3235:
3232:
3229:
3220:
3214:
3195:
3193:
3192:
3187:
3182:
3180:
3172:
3164:
3162:
3158:
3119:
3117:
3109:
3101:
3099:
3095:
3062:
3061:
3039:
3033:
3027:
3023:
3019:
3017:
3016:
3011:
3006:
3002:
2966:
2965:
2946:
2942:
2938:
2911:
2904:
2896:Maclaurin series
2893:
2891:
2890:
2885:
2883:
2846:
2842:
2838:
2831:
2829:
2821:
2820:
2811:
2806:
2804:
2796:
2795:
2786:
2781:
2779:
2771:
2770:
2761:
2742:
2738:
2731:
2729:
2721:
2720:
2711:
2706:
2704:
2696:
2695:
2686:
2681:
2679:
2671:
2670:
2661:
2656:
2654:
2646:
2645:
2636:
2617:
2607:
2605:
2597:
2596:
2587:
2582:
2580:
2572:
2571:
2570:
2557:
2552:
2550:
2542:
2541:
2532:
2527:
2525:
2517:
2516:
2515:
2502:
2497:
2495:
2487:
2486:
2477:
2472:
2470:
2462:
2461:
2460:
2447:
2442:
2440:
2432:
2431:
2422:
2399:
2389:
2387:
2379:
2378:
2377:
2358:
2353:
2351:
2343:
2342:
2341:
2322:
2317:
2315:
2307:
2306:
2305:
2286:
2281:
2279:
2271:
2270:
2269:
2250:
2245:
2243:
2235:
2234:
2233:
2214:
2209:
2207:
2199:
2198:
2197:
2178:
2173:
2171:
2163:
2162:
2161:
2142:
2118:
2117:
2095:
2089:
2087:
2086:
2081:
2079:
2069:
2061:
2053:
2045:
2028:
2027:
2000:
1999:
1975:
1974:
1950:
1949:
1920:
1919:
1892:
1891:
1867:
1866:
1842:
1841:
1821:
1809:
1802:
1785:
1783:
1782:
1777:
1745:
1744:
1725:
1721:
1710:
1703:
1692:
1690:
1689:
1684:
1676:
1672:
1642:
1641:
1620:
1616:
1586:
1585:
1558:
1540:
1536:
1534:
1533:
1528:
1526:
1522:
1492:
1491:
1473:
1471:
1470:
1458:
1432:
1409:
1379:
1371:
1365:
1363:
1362:
1357:
1352:
1351:
1346:
1342:
1341:
1333:
1318:
1300:
1299:
1284:
1277:Limit definition
1272:
1269:for all complex
1268:
1259:has an infinite
1248:
1246:
1245:
1240:
1235:
1233:
1225:
1224:
1215:
1212:
1207:
1183:
1181:
1173:
1172:
1163:
1158:
1156:
1148:
1147:
1138:
1133:
1131:
1120:
1109:
1108:
1093:
1082:
1080:
1079:
1074:
1047:
1045:
1044:
1039:
1031:
1029:
1021:
1013:
1003:complex variable
996:
994:
993:
988:
986:
985:
946:
932:
928:
924:
920:
910:
906:
860:
858:
857:
852:
817:
809:
804:
802:
788:
780:
766:
764:
763:
758:
753:
749:
748:
746:
729:
724:
722:
705:
698:
690:
685:
683:
682:
681:
662:
652:Johann Bernoulli
632:
630:
629:
624:
606:
604:
603:
598:
586:
584:
583:
578:
560:
558:
557:
552:
533:
525:
523:
522:
517:
460:
458:
457:
452:
450:
442:
420:Euler's identity
417:
410:
403:
375:
367:
347:
343:
335:
327:
323:
321:
320:
315:
283:
282:
263:
243:complex analysis
217:
210:
203:
146:
137:
126:
93:Euler's identity
54:
45:
28:
27:
8365:
8364:
8360:
8359:
8358:
8356:
8355:
8354:
8320:
8319:
8318:
8313:
8293:
8254:Euler's theorem
8224:Euler's formula
8165:
8160:
8123:
8105:
8086:
8066:
8064:Further reading
8061:
8051:
8049:
8035:
8031:
8021:
8019:
8011:
8010:
8006:
7998:
7982:
7978:
7971:
7955:
7951:
7939:
7935:
7927:
7913:
7909:
7893:
7889:
7884:
7880:
7864:
7860:
7839:
7835:
7828:
7812:
7805:
7768:
7742:
7741:
7737:
7715:
7701:
7698:
7697:
7672:
7670:
7667:
7666:
7645:
7643:
7640:
7639:
7582:
7569:
7565:
7564:
7542:
7540:
7537:
7536:
7503:
7501:
7498:
7497:
7460:
7446:
7443:
7442:
7417:
7400:
7397:
7396:
7335:
7333:
7330:
7329:
7304:
7287:
7284:
7283:
7244:
7242:
7239:
7238:
7213:
7196:
7193:
7192:
7172:
7168:
7161:
7145:
7141:
7134:
7118:
7114:
7110:
7078:
7062:
7048:
7047:
7029:
7028:
6997:
6988:
6984:
6980:
6978:
6975:
6974:
6963:
6949:
6939:
6927:
6914:
6911:
6906:
6905:
6903:
6902:
6895:
6884:
6875:
6862:
6859:
6854:
6853:
6851:
6850:
6843:
6835:
6826:
6819:
6807:
6739:
6736:
6735:
6731:
6727:
6724:imaginary units
6705:phasor analysis
6682:differentiation
6671:
6670:, the function
6664:
6658:
6616:
6613:
6612:
6596:
6591:
6588:
6587:
6586:of this map is
6562:
6557:
6556:
6554:
6551:
6550:
6530:
6528:
6525:
6524:
6507:
6502:
6501:
6499:
6496:
6495:
6479:
6477:
6474:
6473:
6441:
6437:
6429:
6426:
6425:
6414:
6403:
6397:
6394:
6379:
6363:
6352:
6347:
6330:
6323:
6319:
6312:
6296:
6295:
6199:
6198:
6169:
6165:
6123:
6119:
6118:
6114:
6099:
6098:
6092:
6091:
6085:
6084:
6072:
6068:
6050:
6032:
6028:
6016:
6012:
6011:
6009:
6002:
6001:
5974:
5970:
5964:
5963:
5948:
5947:
5933:
5929:
5902:
5898:
5897:
5893:
5878:
5877:
5861:
5857:
5853:
5840:
5824:
5822:
5819:
5818:
5798:
5797:
5743:
5739:
5729:
5720:
5719:
5713:
5712:
5682:
5678:
5654:
5650:
5649:
5647:
5614:
5610:
5586:
5582:
5581:
5579:
5573:
5572:
5562:
5553:
5552:
5522:
5518:
5491:
5487:
5463:
5459:
5435:
5431:
5430:
5428:
5415:
5406:
5405:
5387:
5383:
5371:
5367:
5366:
5364:
5343:
5339:
5327:
5323:
5322:
5320:
5313:
5291:
5289:
5286:
5285:
5262:
5261:
5225:
5221:
5209:
5205:
5204:
5202:
5195:
5180:
5179:
5146:
5142:
5130:
5126:
5125:
5123:
5116:
5100:
5098:
5095:
5094:
5078:
5077:
5041:
5037:
5028:
5024:
5023:
5021:
5008:
5001:
4997:
4985:
4981:
4980:
4978:
4971:
4956:
4955:
4928:
4924:
4912:
4908:
4907:
4905:
4898:
4882:
4880:
4877:
4876:
4866:
4862:
4845:
4844:
4768:
4756:
4752:
4749:
4748:
4714:
4705:
4701:
4697:
4695:
4692:
4691:
4675:
4674:
4661:
4648:
4644:
4632:
4628:
4627:
4625:
4609:
4605:
4601:
4588:
4576:
4575:
4554:
4550:
4538:
4534:
4533:
4531:
4515:
4511:
4507:
4494:
4481:
4479:
4476:
4475:
4451:
4434:
4411:
4407:
4398:
4388:
4384:
4380:
4379:
4377:
4374:
4373:
4366:
4324:
4304:
4301:
4300:
4293:
4260:
4253:
4249:
4237:
4233:
4220:
4213:
4209:
4197:
4193:
4182:
4174:
4171:
4170:
4166:
4162:
4136:
4132:
4123:
4119:
4113:
4109:
4107:
4104:
4103:
4077:
4073:
4065:
4062:
4061:
4054:
4039:
4036:
4031:
4030:
4028:
4022:
4018:
4006:
3994:
3990:
3986:
3979:
3964:
3955:
3952:
3949:
3948:
3946:
3940:
3933:
3913:
3905:
3879:
3873:
3857:
3855:
3849:
3844:
3832:
3820:
3803:
3802:
3787:
3783:
3742:
3734:
3712:
3701:
3700:
3697:
3696:
3684:
3680:
3639:
3631:
3609:
3602:
3600:
3597:
3596:
3581:
3580:
3566:
3553:A point in the
3547:
3535:
3528:
3521:
3517:
3511:
3485:
3481:
3463:
3460:
3436:
3429:
3424:
3334:
3330:
3328:
3325:
3324:
3314:
3307:
3300:
3293:
3286:
3282:
3272:
3268:
3264:
3254:
3251:
3248:
3247:
3245:
3243:
3233:
3230:
3227:
3226:
3224:
3222:
3216:
3197:
3173:
3165:
3163:
3130:
3126:
3110:
3102:
3100:
3070:
3066:
3054:
3050:
3045:
3042:
3041:
3035:
3029:
3025:
3021:
2977:
2973:
2958:
2954:
2952:
2949:
2948:
2944:
2940:
2936:
2926:
2906:
2899:
2881:
2880:
2844:
2843:
2822:
2816:
2812:
2810:
2797:
2791:
2787:
2785:
2772:
2766:
2762:
2760:
2753:
2749:
2722:
2716:
2712:
2710:
2697:
2691:
2687:
2685:
2672:
2666:
2662:
2660:
2647:
2641:
2637:
2635:
2628:
2624:
2615:
2614:
2598:
2592:
2588:
2586:
2573:
2566:
2562:
2558:
2556:
2543:
2537:
2533:
2531:
2518:
2511:
2507:
2503:
2501:
2488:
2482:
2478:
2476:
2463:
2456:
2452:
2448:
2446:
2433:
2427:
2423:
2421:
2397:
2396:
2380:
2373:
2369:
2359:
2357:
2344:
2337:
2333:
2323:
2321:
2308:
2301:
2297:
2287:
2285:
2272:
2265:
2261:
2251:
2249:
2236:
2229:
2225:
2215:
2213:
2200:
2193:
2189:
2179:
2177:
2164:
2157:
2153:
2143:
2141:
2119:
2110:
2106:
2102:
2100:
2097:
2096:
2093:
2077:
2076:
2068:
2060:
2052:
2043:
2042:
2029:
2023:
2019:
2017:
2001:
1995:
1991:
1989:
1976:
1970:
1966:
1964:
1951:
1945:
1941:
1938:
1937:
1921:
1915:
1911:
1909:
1893:
1887:
1883:
1881:
1868:
1862:
1858:
1856:
1843:
1837:
1833:
1829:
1827:
1824:
1823:
1819:
1804:
1798:
1791:
1737:
1733:
1731:
1728:
1727:
1723:
1712:
1705:
1694:
1647:
1643:
1631:
1627:
1591:
1587:
1575:
1571:
1551:
1549:
1546:
1545:
1538:
1497:
1493:
1481:
1477:
1463:
1459:
1433:
1431:
1414:
1411:
1410:
1400:
1394:
1386:
1377:
1369:
1347:
1332:
1325:
1321:
1320:
1308:
1295:
1291:
1289:
1286:
1285:
1282:
1279:
1270:
1264:
1263:and so defines
1226:
1220:
1216:
1214:
1208:
1197:
1174:
1168:
1164:
1162:
1149:
1143:
1139:
1137:
1124:
1119:
1104:
1100:
1098:
1095:
1094:
1091:
1088:
1053:
1050:
1049:
1022:
1014:
1012:
1010:
1007:
1006:
981:
977:
960:
957:
956:
953:
942:
930:
926:
922:
916:
908:
902:
899:
889:
808:
789:
781:
779:
774:
771:
770:
733:
728:
709:
704:
703:
699:
689:
677:
673:
666:
661:
659:
656:
655:
654:had found that
612:
609:
608:
592:
589:
588:
566:
563:
562:
546:
543:
542:
528:
466:
463:
462:
441:
439:
436:
435:
428:
412:
405:
395:
389:Richard Feynman
382:Euler's formula
373:
360:
345:
341:
333:
325:
275:
271:
269:
266:
265:
261:
228:Euler's formula
221:
174:
144:
135:
124:
98:Euler's formula
41:
24:
17:
12:
11:
5:
8363:
8353:
8352:
8350:Leonhard Euler
8347:
8342:
8337:
8332:
8315:
8314:
8312:
8311:
8298:
8295:
8294:
8292:
8291:
8286:
8281:
8276:
8271:
8266:
8264:Euler function
8261:
8256:
8251:
8246:
8241:
8236:
8231:
8226:
8221:
8216:
8211:
8206:
8201:
8196:
8191:
8186:
8181:
8176:
8170:
8167:
8166:
8163:Leonhard Euler
8159:
8158:
8151:
8144:
8136:
8130:
8129:
8122:
8121:External links
8119:
8118:
8117:
8103:
8090:
8084:
8065:
8062:
8060:
8059:
8038:Hartl, Michael
8029:
8004:
7996:
7976:
7969:
7949:
7933:
7926:978-0201002881
7925:
7907:
7887:
7878:
7866:Leonhard Euler
7858:
7833:
7826:
7803:
7801:
7800:
7795:
7784:
7754:
7749:
7746:
7740:
7736:
7733:
7730:
7727:
7722:
7719:
7714:
7711:
7708:
7705:
7679:
7676:
7652:
7649:
7627:
7624:
7621:
7618:
7615:
7612:
7604:
7600:
7597:
7594:
7589:
7586:
7581:
7578:
7575:
7572:
7568:
7563:
7560:
7557:
7554:
7549:
7546:
7510:
7507:
7478:
7475:
7472:
7467:
7464:
7459:
7456:
7453:
7450:
7424:
7421:
7416:
7413:
7410:
7407:
7404:
7387:and the point
7342:
7339:
7328:multiplied by
7311:
7308:
7303:
7300:
7297:
7294:
7291:
7251:
7248:
7220:
7217:
7212:
7209:
7206:
7203:
7200:
7173:Cotes wrote:
7166:
7159:
7139:
7132:
7111:
7109:
7106:
7105:
7104:
7099:
7094:
7089:
7084:
7082:Complex number
7077:
7074:
7046:
7043:
7040:
7037:
7034:
7032:
7030:
7027:
7024:
7021:
7018:
7015:
7012:
7009:
7006:
7003:
7000:
6998:
6994:
6991:
6987:
6983:
6982:
6934:
6933:
6925:
6899:
6898:
6893:
6881:
6880:
6873:
6847:
6846:
6841:
6832:
6831:
6824:
6806:
6803:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6657:
6654:
6629:
6626:
6623:
6620:
6599:
6595:
6582:says that the
6565:
6560:
6547:covering space
6533:
6510:
6505:
6482:
6447:
6444:
6440:
6436:
6433:
6416:
6415:
6366:
6364:
6357:
6351:
6348:
6326:(in radians).
6322:and arbitrary
6294:
6291:
6288:
6285:
6282:
6279:
6276:
6273:
6270:
6267:
6264:
6261:
6258:
6255:
6252:
6249:
6246:
6243:
6240:
6237:
6234:
6231:
6228:
6225:
6222:
6219:
6216:
6213:
6210:
6207:
6204:
6202:
6200:
6196:
6190:
6187:
6184:
6181:
6178:
6175:
6172:
6168:
6164:
6161:
6158:
6155:
6152:
6149:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6122:
6117:
6113:
6110:
6107:
6104:
6102:
6100:
6095:
6088:
6081:
6078:
6075:
6071:
6067:
6062:
6059:
6056:
6053:
6047:
6041:
6038:
6035:
6031:
6027:
6022:
6019:
6015:
6005:
6000:
5995:
5992:
5989:
5986:
5983:
5980:
5977:
5973:
5967:
5962:
5959:
5956:
5953:
5951:
5949:
5945:
5939:
5936:
5932:
5928:
5923:
5920:
5917:
5914:
5911:
5908:
5905:
5901:
5896:
5892:
5889:
5886:
5883:
5881:
5879:
5875:
5870:
5867:
5864:
5860:
5856:
5852:
5849:
5846:
5843:
5841:
5839:
5836:
5833:
5830:
5827:
5826:
5796:
5792:
5788:
5785:
5782:
5779:
5776:
5773:
5770:
5767:
5764:
5761:
5758:
5755:
5752:
5749:
5746:
5742:
5736:
5733:
5728:
5725:
5723:
5721:
5716:
5709:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5681:
5677:
5672:
5669:
5666:
5663:
5660:
5657:
5653:
5646:
5641:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5613:
5609:
5604:
5601:
5598:
5595:
5592:
5589:
5585:
5576:
5569:
5566:
5561:
5558:
5556:
5554:
5549:
5543:
5540:
5537:
5534:
5531:
5528:
5525:
5521:
5517:
5512:
5509:
5506:
5503:
5500:
5497:
5494:
5490:
5486:
5481:
5478:
5475:
5472:
5469:
5466:
5462:
5458:
5453:
5450:
5447:
5444:
5441:
5438:
5434:
5427:
5422:
5419:
5414:
5411:
5409:
5407:
5402:
5396:
5393:
5390:
5386:
5382:
5377:
5374:
5370:
5363:
5358:
5352:
5349:
5346:
5342:
5338:
5333:
5330:
5326:
5319:
5316:
5314:
5312:
5309:
5306:
5303:
5300:
5297:
5294:
5293:
5260:
5257:
5254:
5251:
5248:
5245:
5240:
5234:
5231:
5228:
5224:
5220:
5215:
5212:
5208:
5201:
5198:
5196:
5194:
5191:
5188:
5185:
5182:
5181:
5178:
5175:
5172:
5169:
5166:
5161:
5155:
5152:
5149:
5145:
5141:
5136:
5133:
5129:
5122:
5119:
5117:
5115:
5112:
5109:
5106:
5103:
5102:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5053:
5047:
5044:
5040:
5036:
5031:
5027:
5020:
5014:
5011:
5004:
5000:
4996:
4991:
4988:
4984:
4977:
4974:
4972:
4970:
4967:
4964:
4961:
4958:
4957:
4954:
4951:
4948:
4945:
4942:
4937:
4931:
4927:
4923:
4918:
4915:
4911:
4904:
4901:
4899:
4897:
4894:
4891:
4888:
4885:
4884:
4843:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4769:
4765:
4762:
4759:
4755:
4751:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4715:
4711:
4708:
4704:
4700:
4699:
4673:
4667:
4664:
4657:
4654:
4651:
4647:
4643:
4638:
4635:
4631:
4624:
4620:
4615:
4612:
4608:
4604:
4600:
4597:
4594:
4591:
4589:
4587:
4584:
4581:
4578:
4577:
4574:
4569:
4563:
4560:
4557:
4553:
4549:
4544:
4541:
4537:
4530:
4526:
4521:
4518:
4514:
4510:
4506:
4503:
4500:
4497:
4495:
4493:
4490:
4487:
4484:
4483:
4450:
4447:
4422:
4417:
4414:
4410:
4406:
4401:
4396:
4391:
4387:
4383:
4346:
4343:
4340:
4337:
4333:
4330:
4327:
4323:
4320:
4317:
4314:
4311:
4308:
4279:
4276:
4273:
4269:
4266:
4263:
4259:
4256:
4252:
4248:
4243:
4240:
4236:
4229:
4226:
4223:
4219:
4216:
4212:
4208:
4203:
4200:
4196:
4191:
4188:
4185:
4181:
4178:
4150:
4145:
4142:
4139:
4135:
4131:
4126:
4122:
4116:
4112:
4091:
4086:
4083:
4080:
4076:
4072:
4069:
4053:
4050:
3903:
3902:
3877:
3842:
3830:
3801:
3796:
3793:
3790:
3786:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3745:
3741:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3713:
3708:
3705:
3699:
3698:
3695:
3690:
3687:
3683:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3642:
3638:
3634:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3610:
3608:
3605:
3604:
3459:
3456:
3428:
3425:
3423:
3420:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3340:
3337:
3333:
3185:
3179:
3176:
3171:
3168:
3161:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3129:
3125:
3122:
3116:
3113:
3108:
3105:
3098:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3069:
3065:
3060:
3057:
3053:
3049:
3009:
3005:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2976:
2972:
2969:
2964:
2961:
2957:
2925:
2922:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2847:
2845:
2841:
2837:
2834:
2828:
2825:
2819:
2815:
2809:
2803:
2800:
2794:
2790:
2784:
2778:
2775:
2769:
2765:
2759:
2756:
2752:
2748:
2745:
2741:
2737:
2734:
2728:
2725:
2719:
2715:
2709:
2703:
2700:
2694:
2690:
2684:
2678:
2675:
2669:
2665:
2659:
2653:
2650:
2644:
2640:
2634:
2631:
2627:
2623:
2620:
2618:
2616:
2613:
2610:
2604:
2601:
2595:
2591:
2585:
2579:
2576:
2569:
2565:
2561:
2555:
2549:
2546:
2540:
2536:
2530:
2524:
2521:
2514:
2510:
2506:
2500:
2494:
2491:
2485:
2481:
2475:
2469:
2466:
2459:
2455:
2451:
2445:
2439:
2436:
2430:
2426:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2400:
2398:
2395:
2392:
2386:
2383:
2376:
2372:
2368:
2365:
2362:
2356:
2350:
2347:
2340:
2336:
2332:
2329:
2326:
2320:
2314:
2311:
2304:
2300:
2296:
2293:
2290:
2284:
2278:
2275:
2268:
2264:
2260:
2257:
2254:
2248:
2242:
2239:
2232:
2228:
2224:
2221:
2218:
2212:
2206:
2203:
2196:
2192:
2188:
2185:
2182:
2176:
2170:
2167:
2160:
2156:
2152:
2149:
2146:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2120:
2116:
2113:
2109:
2105:
2104:
2075:
2072:
2070:
2067:
2064:
2062:
2059:
2056:
2054:
2051:
2048:
2046:
2044:
2041:
2038:
2035:
2032:
2030:
2026:
2022:
2018:
2016:
2013:
2010:
2007:
2004:
2002:
1998:
1994:
1990:
1988:
1985:
1982:
1979:
1977:
1973:
1969:
1965:
1963:
1960:
1957:
1954:
1952:
1948:
1944:
1940:
1939:
1936:
1933:
1930:
1927:
1924:
1922:
1918:
1914:
1910:
1908:
1905:
1902:
1899:
1896:
1894:
1890:
1886:
1882:
1880:
1877:
1874:
1871:
1869:
1865:
1861:
1857:
1855:
1852:
1849:
1846:
1844:
1840:
1836:
1832:
1831:
1790:
1787:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1743:
1740:
1736:
1682:
1679:
1675:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1646:
1640:
1637:
1634:
1630:
1626:
1623:
1619:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1590:
1584:
1581:
1578:
1574:
1570:
1567:
1564:
1561:
1557:
1554:
1525:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1496:
1490:
1487:
1484:
1480:
1476:
1469:
1466:
1462:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1430:
1427:
1424:
1421:
1418:
1393:
1390:
1385:
1382:
1355:
1350:
1345:
1339:
1336:
1331:
1328:
1324:
1317:
1314:
1311:
1307:
1303:
1298:
1294:
1278:
1275:
1238:
1232:
1229:
1223:
1219:
1211:
1206:
1203:
1200:
1196:
1192:
1189:
1186:
1180:
1177:
1171:
1167:
1161:
1155:
1152:
1146:
1142:
1136:
1130:
1127:
1123:
1118:
1115:
1112:
1107:
1103:
1087:
1084:
1072:
1069:
1066:
1063:
1060:
1057:
1037:
1034:
1028:
1025:
1020:
1017:
997:is the unique
984:
980:
976:
973:
970:
967:
964:
952:
949:
888:
885:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
815:
812:
807:
801:
798:
795:
792:
787:
784:
778:
756:
752:
745:
742:
739:
736:
732:
727:
721:
718:
715:
712:
708:
702:
696:
693:
688:
680:
676:
672:
669:
665:
639:Leonhard Euler
622:
619:
616:
596:
576:
573:
570:
550:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
448:
445:
427:
424:
378:complex number
368:("cosine plus
338:imaginary unit
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
281:
278:
274:
232:Leonhard Euler
230:, named after
223:
222:
220:
219:
212:
205:
197:
194:
193:
192:
191:
183:
182:
181:Related topics
178:
177:
176:
175:
172:Leonhard Euler
169:
161:
160:
156:
155:
154:
153:
148:
140:
128:
127:
120:
119:
118:
117:
116:
115:
100:
95:
90:
82:
81:
77:
76:
75:
74:
69:
61:
60:
56:
55:
47:
46:
37:
36:
15:
9:
6:
4:
3:
2:
8362:
8351:
8348:
8346:
8343:
8341:
8338:
8336:
8333:
8331:
8328:
8327:
8325:
8310:
8309:
8300:
8299:
8296:
8290:
8287:
8285:
8282:
8280:
8277:
8275:
8274:Euler numbers
8272:
8270:
8267:
8265:
8262:
8260:
8257:
8255:
8252:
8250:
8247:
8245:
8242:
8240:
8237:
8235:
8232:
8230:
8227:
8225:
8222:
8220:
8217:
8215:
8212:
8210:
8207:
8205:
8202:
8200:
8197:
8195:
8192:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8171:
8168:
8164:
8157:
8152:
8150:
8145:
8143:
8138:
8137:
8134:
8128:
8125:
8124:
8114:
8110:
8106:
8100:
8096:
8091:
8087:
8081:
8076:
8075:
8068:
8067:
8047:
8043:
8039:
8033:
8018:
8014:
8008:
7999:
7997:0-9614088-2-0
7993:
7989:
7988:
7980:
7972:
7970:9780123859136
7966:
7962:
7961:
7953:
7947:
7943:
7937:
7928:
7922:
7918:
7911:
7903:
7899:
7891:
7882:
7875:
7871:
7867:
7862:
7856:
7852:
7849:
7845:
7844:
7837:
7829:
7827:9781441960528
7823:
7819:
7818:
7810:
7808:
7799:
7796:
7793:
7789:
7785:
7783:
7779:
7775:
7771:
7770:
7752:
7747:
7744:
7738:
7734:
7731:
7728:
7725:
7720:
7717:
7712:
7709:
7706:
7703:
7695:
7677:
7674:
7650:
7647:
7625:
7619:
7616:
7610:
7602:
7598:
7595:
7592:
7587:
7584:
7579:
7576:
7573:
7570:
7566:
7561:
7558:
7555:
7552:
7547:
7544:
7534:
7530:
7526:
7508:
7505:
7495:
7476:
7473:
7470:
7465:
7462:
7457:
7454:
7451:
7448:
7440:
7422:
7419:
7414:
7411:
7408:
7405:
7402:
7394:
7390:
7386:
7382:
7378:
7374:
7370:
7366:
7362:
7358:
7340:
7337:
7327:
7309:
7306:
7301:
7298:
7295:
7292:
7289:
7281:
7277:
7273:
7269:
7265:
7249:
7246:
7234:
7218:
7215:
7210:
7207:
7204:
7201:
7198:
7188:
7184:
7180:
7176:
7170:
7162:
7160:0-201-02010-6
7156:
7152:
7151:
7143:
7135:
7133:981-02-4780-X
7129:
7125:
7124:
7116:
7112:
7103:
7100:
7098:
7095:
7093:
7090:
7088:
7085:
7083:
7080:
7079:
7073:
7071:
7065:
7044:
7041:
7038:
7035:
7033:
7025:
7022:
7019:
7016:
7013:
7010:
7007:
7004:
7001:
6999:
6992:
6989:
6985:
6972:
6966:
6961:
6956:
6952:
6946:
6942:
6931:
6926:
6923:
6909:
6901:
6900:
6894:
6891:
6887:
6883:
6882:
6878:
6874:
6871:
6857:
6849:
6848:
6842:
6839:
6834:
6833:
6829:
6825:
6822:
6818:
6817:
6814:
6812:
6811:special cases
6802:
6800:
6796:
6780:
6777:
6774:
6771:
6768:
6765:
6762:
6759:
6756:
6753:
6750:
6747:
6744:
6741:
6725:
6721:
6718:, there is a
6717:
6713:
6708:
6706:
6702:
6698:
6694:
6690:
6685:
6683:
6679:
6678:eigenfunction
6674:
6669:
6663:
6649:
6645:
6643:
6627:
6624:
6621:
6618:
6593:
6585:
6581:
6578:. Similarly,
6563:
6548:
6508:
6471:
6467:
6463:
6445:
6442:
6438:
6431:
6423:
6412:
6409:
6401:
6398:November 2022
6391:
6387:
6383:
6377:
6376:
6372:
6367:This section
6365:
6361:
6356:
6355:
6346:
6341:
6339:
6334:
6327:
6316:
6309:
6292:
6286:
6280:
6277:
6274:
6265:
6262:
6259:
6253:
6250:
6247:
6244:
6238:
6232:
6226:
6223:
6220:
6211:
6208:
6205:
6203:
6194:
6188:
6182:
6179:
6176:
6170:
6166:
6162:
6159:
6156:
6153:
6150:
6147:
6142:
6136:
6133:
6130:
6124:
6120:
6115:
6111:
6108:
6105:
6103:
6079:
6076:
6073:
6069:
6065:
6060:
6057:
6054:
6051:
6045:
6039:
6036:
6033:
6029:
6025:
6020:
6017:
6013:
5998:
5993:
5987:
5984:
5981:
5975:
5971:
5960:
5957:
5954:
5952:
5943:
5937:
5934:
5930:
5926:
5921:
5915:
5912:
5909:
5903:
5899:
5894:
5890:
5887:
5884:
5882:
5873:
5868:
5865:
5862:
5858:
5854:
5850:
5847:
5844:
5842:
5837:
5834:
5831:
5828:
5816:
5811:
5794:
5790:
5783:
5780:
5777:
5771:
5768:
5765:
5759:
5756:
5753:
5747:
5744:
5740:
5734:
5731:
5726:
5724:
5707:
5698:
5695:
5692:
5686:
5683:
5679:
5675:
5667:
5664:
5661:
5655:
5651:
5644:
5639:
5630:
5627:
5624:
5618:
5615:
5611:
5607:
5599:
5596:
5593:
5587:
5583:
5567:
5564:
5559:
5557:
5547:
5538:
5535:
5532:
5529:
5523:
5519:
5515:
5507:
5504:
5501:
5498:
5492:
5488:
5484:
5476:
5473:
5470:
5464:
5460:
5456:
5448:
5445:
5442:
5436:
5432:
5425:
5420:
5417:
5412:
5410:
5400:
5394:
5391:
5388:
5384:
5380:
5375:
5372:
5368:
5361:
5356:
5350:
5347:
5344:
5340:
5336:
5331:
5328:
5324:
5317:
5315:
5310:
5307:
5304:
5301:
5298:
5295:
5283:
5281:
5275:
5258:
5255:
5252:
5249:
5246:
5243:
5238:
5232:
5229:
5226:
5222:
5218:
5213:
5210:
5206:
5199:
5197:
5192:
5189:
5186:
5183:
5176:
5173:
5170:
5167:
5164:
5159:
5153:
5150:
5147:
5143:
5139:
5134:
5131:
5127:
5120:
5118:
5113:
5110:
5107:
5104:
5091:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5051:
5045:
5042:
5038:
5034:
5029:
5025:
5018:
5012:
5009:
5002:
4998:
4994:
4989:
4986:
4982:
4975:
4973:
4968:
4965:
4962:
4959:
4952:
4949:
4946:
4943:
4940:
4935:
4929:
4925:
4921:
4916:
4913:
4909:
4902:
4900:
4895:
4892:
4889:
4886:
4873:
4869:
4859:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4811:
4808:
4802:
4799:
4796:
4793:
4787:
4784:
4778:
4775:
4772:
4770:
4763:
4760:
4757:
4753:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4716:
4709:
4706:
4702:
4688:
4671:
4665:
4662:
4655:
4652:
4649:
4645:
4641:
4636:
4633:
4629:
4622:
4618:
4613:
4610:
4606:
4602:
4598:
4595:
4592:
4590:
4585:
4582:
4579:
4572:
4567:
4561:
4558:
4555:
4551:
4547:
4542:
4539:
4535:
4528:
4524:
4519:
4516:
4512:
4508:
4504:
4501:
4498:
4496:
4491:
4488:
4485:
4473:
4472:weighted sums
4469:
4465:
4455:
4446:
4444:
4441:, as well as
4440:
4420:
4415:
4412:
4408:
4404:
4399:
4394:
4389:
4385:
4381:
4370:
4364:
4360:
4344:
4341:
4338:
4335:
4331:
4328:
4325:
4321:
4318:
4315:
4312:
4309:
4306:
4296:
4277:
4274:
4271:
4267:
4264:
4261:
4257:
4254:
4250:
4246:
4241:
4238:
4234:
4227:
4224:
4221:
4217:
4214:
4210:
4206:
4201:
4198:
4194:
4189:
4186:
4183:
4179:
4176:
4148:
4143:
4140:
4137:
4133:
4129:
4124:
4120:
4114:
4110:
4089:
4084:
4081:
4078:
4074:
4070:
4067:
4059:
4049:
4042:
4034:
4026:
4014:
4010:
4002:
3998:
3982:
3975:
3971:
3967:
3961:
3943:
3937:
3931:
3927:
3923:
3919:
3911:
3898:
3894:
3890:
3886:
3882:
3878:
3871:
3864:
3860:
3852:
3847:
3843:
3839:
3835:
3831:
3827:
3823:
3819:
3818:
3817:
3799:
3794:
3791:
3788:
3784:
3780:
3777:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3739:
3731:
3728:
3725:
3722:
3719:
3716:
3714:
3703:
3693:
3688:
3685:
3681:
3677:
3674:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3636:
3628:
3625:
3622:
3619:
3616:
3613:
3611:
3606:
3593:
3589:
3584:
3577:
3573:
3569:
3564:
3560:
3556:
3555:complex plane
3551:
3546:numbers
3545:
3541:
3532:
3525:
3514:
3510:
3506:
3505:Taylor series
3501:
3499:
3495:
3491:
3479:
3478:complex plane
3475:
3471:
3466:
3451:
3447:
3443:
3439:
3433:
3419:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3346:
3343:
3338:
3335:
3331:
3321:
3317:
3310:
3303:
3296:
3289:
3279:
3275:
3260:
3239:
3219:
3212:
3208:
3204:
3200:
3196:Substituting
3183:
3177:
3174:
3169:
3166:
3159:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3127:
3123:
3120:
3114:
3111:
3106:
3103:
3096:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3067:
3063:
3058:
3055:
3051:
3047:
3038:
3032:
3007:
3003:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2974:
2970:
2967:
2962:
2959:
2955:
2943:depending on
2935:
2932:. Therefore,
2931:
2921:
2919:
2915:
2910:
2903:
2897:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2848:
2839:
2835:
2832:
2826:
2823:
2817:
2813:
2807:
2801:
2798:
2792:
2788:
2782:
2776:
2773:
2767:
2763:
2757:
2754:
2750:
2746:
2743:
2739:
2735:
2732:
2726:
2723:
2717:
2713:
2707:
2701:
2698:
2692:
2688:
2682:
2676:
2673:
2667:
2663:
2657:
2651:
2648:
2642:
2638:
2632:
2629:
2625:
2621:
2619:
2611:
2608:
2602:
2599:
2593:
2589:
2583:
2577:
2574:
2567:
2563:
2559:
2553:
2547:
2544:
2538:
2534:
2528:
2522:
2519:
2512:
2508:
2504:
2498:
2492:
2489:
2483:
2479:
2473:
2467:
2464:
2457:
2453:
2449:
2443:
2437:
2434:
2428:
2424:
2418:
2415:
2412:
2409:
2406:
2403:
2401:
2393:
2390:
2384:
2381:
2374:
2366:
2363:
2354:
2348:
2345:
2338:
2330:
2327:
2318:
2312:
2309:
2302:
2294:
2291:
2282:
2276:
2273:
2266:
2258:
2255:
2246:
2240:
2237:
2230:
2222:
2219:
2210:
2204:
2201:
2194:
2186:
2183:
2174:
2168:
2165:
2158:
2150:
2147:
2138:
2135:
2132:
2129:
2126:
2123:
2121:
2114:
2111:
2107:
2090:
2073:
2071:
2065:
2063:
2057:
2055:
2049:
2047:
2039:
2036:
2033:
2031:
2024:
2020:
2014:
2011:
2008:
2005:
2003:
1996:
1992:
1986:
1983:
1980:
1978:
1971:
1967:
1961:
1958:
1955:
1953:
1946:
1942:
1934:
1931:
1928:
1925:
1923:
1916:
1912:
1906:
1903:
1900:
1897:
1895:
1888:
1884:
1878:
1875:
1872:
1870:
1863:
1859:
1853:
1850:
1847:
1845:
1838:
1834:
1817:
1807:
1801:
1795:
1786:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1741:
1738:
1734:
1722:for all real
1719:
1715:
1708:
1701:
1697:
1680:
1677:
1673:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1644:
1638:
1635:
1632:
1628:
1624:
1621:
1617:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1588:
1582:
1579:
1576:
1572:
1568:
1562:
1555:
1552:
1544:
1523:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1494:
1488:
1485:
1482:
1478:
1474:
1467:
1464:
1460:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1428:
1422:
1416:
1407:
1403:
1397:
1389:
1381:
1375:
1366:
1353:
1348:
1343:
1337:
1334:
1329:
1326:
1322:
1309:
1301:
1296:
1292:
1274:
1267:
1262:
1258:
1254:
1249:
1236:
1230:
1227:
1221:
1217:
1204:
1201:
1198:
1194:
1190:
1187:
1184:
1178:
1175:
1169:
1165:
1159:
1153:
1150:
1144:
1140:
1134:
1128:
1125:
1121:
1116:
1113:
1110:
1105:
1101:
1083:
1070:
1067:
1061:
1055:
1035:
1032:
1026:
1023:
1018:
1015:
1004:
1000:
982:
978:
974:
968:
962:
948:
945:
940:
937:
919:
914:
905:
898:
894:
884:
882:
881:Caspar Wessel
878:
877:complex plane
873:
871:
866:
864:
848:
845:
842:
836:
833:
830:
827:
821:
818:
813:
810:
805:
799:
796:
793:
790:
785:
782:
776:
767:
754:
750:
743:
740:
737:
734:
730:
725:
719:
716:
713:
710:
706:
700:
694:
691:
686:
678:
674:
670:
667:
663:
653:
649:
647:
646:
640:
620:
617:
614:
594:
574:
571:
568:
548:
539:
535:
532:
513:
507:
504:
501:
498:
495:
492:
489:
486:
480:
477:
474:
471:
468:
446:
443:
433:
423:
421:
415:
408:
402:
398:
392:
390:
385:
383:
379:
371:
366:
363:
358:
354:
351:
339:
331:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
279:
276:
272:
259:
255:
252:
248:
244:
240:
237:
233:
229:
218:
213:
211:
206:
204:
199:
198:
196:
195:
190:
187:
186:
185:
184:
180:
179:
173:
170:
168:
165:
164:
163:
162:
158:
157:
152:
149:
147:
141:
139:
138:is irrational
132:
131:
130:
129:
122:
121:
114:
110:
106:
105:
104:
101:
99:
96:
94:
91:
89:
86:
85:
84:
83:
79:
78:
73:
70:
68:
65:
64:
63:
62:
58:
57:
53:
49:
48:
44:
39:
38:
34:
30:
29:
26:
22:
8345:Trigonometry
8306:
8269:Euler method
8223:
8094:
8073:
8052:14 September
8050:. Retrieved
8032:
8022:10 September
8020:. Retrieved
8016:
8007:
7986:
7979:
7959:
7952:
7936:
7931:Theorem 1.42
7916:
7910:
7901:
7897:
7890:
7881:
7861:
7841:
7836:
7820:. Springer.
7816:
7787:
7777:
7773:
7693:
7532:
7528:
7524:
7493:
7438:
7392:
7388:
7384:
7380:
7376:
7372:
7368:
7364:
7360:
7356:
7325:
7279:
7275:
7271:
7270:, has sinus
7267:
7236:
7190:
7186:
7182:
7178:
7174:
7169:
7149:
7142:
7122:
7115:
7063:
6964:
6954:
6950:
6944:
6940:
6937:
6929:
6921:
6907:
6889:
6885:
6876:
6869:
6855:
6837:
6827:
6820:
6808:
6709:
6686:
6672:
6665:
6419:
6404:
6395:
6380:Please help
6368:
6332:
6329:Considering
6328:
6314:
6310:
5812:
5284:
5279:
5276:
5093:In addition
5092:
4871:
4867:
4860:
4689:
4468:trigonometry
4460:
4371:
4294:
4055:
4040:
4032:
4024:
4012:
4008:
4000:
3996:
3980:
3973:
3969:
3965:
3945:
3941:
3935:
3932:addition of
3921:
3917:
3904:
3896:
3892:
3884:
3880:
3862:
3858:
3850:
3845:
3837:
3833:
3825:
3821:
3591:
3587:
3582:
3575:
3571:
3567:
3552:
3543:
3530:
3523:
3512:
3502:
3464:
3461:
3449:
3445:
3441:
3437:
3422:Applications
3319:
3315:
3308:
3301:
3294:
3287:
3277:
3273:
3244:
3223:
3217:
3210:
3206:
3202:
3198:
3036:
3030:
2927:
2908:
2901:
2091:
1813:
1805:
1799:
1726:, and thus
1717:
1713:
1706:
1699:
1695:
1543:product rule
1405:
1401:
1398:
1395:
1387:
1367:
1281:For complex
1280:
1265:
1257:power series
1250:
1090:For complex
1089:
954:
943:
929:in place of
917:
903:
900:
874:
867:
768:
650:
643:
637:Around 1740
636:
530:
429:
413:
406:
400:
396:
393:
386:
381:
369:
364:
236:mathematical
227:
226:
107:exponential
97:
80:Applications
25:
7940:user02138 (
7782:Hathi Trust
6716:quaternions
4875:, we have:
3963:instead of
3474:unit circle
432:Roger Cotes
258:real number
167:John Napier
134:proof that
8324:Categories
7904:: 289–297.
7108:References
6660:See also:
6462:surjective
6343:See also:
4365:, because
4017:differ by
3299:come from
1253:ratio test
1251:Using the
769:And since
264:, one has
103:half-lives
59:Properties
8289:Namesakes
7753:θ
7745:−
7732:θ
7729:
7718:−
7710:θ
7707:
7675:−
7648:−
7626:θ
7599:θ
7596:
7585:−
7577:θ
7574:
7562:
7545:−
7535:. Thus,
7506:−
7477:θ
7474:
7463:−
7455:θ
7452:
7420:−
7338:−
7307:−
7247:−
7216:−
7026:τ
7023:
7011:τ
7008:
6993:τ
6962:) yields
6775:
6760:
6745:
6701:imaginary
6628:π
6619:τ
6594:τ
6435:↦
6369:does not
6278:−
6266:
6260:−
6251:
6239:⋅
6224:−
6212:
6180:−
6163:−
6157:
6148:⋅
6134:−
6112:
6074:−
6066:−
6058:
6046:⏟
6034:−
5999:⋅
5985:−
5961:
5927:⋅
5913:−
5891:
5851:
5832:
5815:real part
5781:−
5772:
5748:
5696:−
5684:−
5665:−
5616:−
5536:−
5530:−
5499:−
5474:−
5426:⋅
5389:−
5362:⋅
5345:−
5308:
5299:
5253:
5227:−
5219:−
5187:
5171:
5148:−
5108:
5069:
5043:−
5035:−
4995:−
4987:−
4963:
4947:
4914:−
4890:
4839:
4830:−
4824:
4809:−
4803:
4785:−
4779:
4758:−
4740:
4725:
4650:−
4642:−
4599:
4583:
4556:−
4505:
4489:
4342:φ
4322:
4310:
4278:φ
4258:
4242:φ
4218:
4202:φ
4102:and that
4082:
4058:logarithm
3870:magnitude
3854:| =
3795:φ
3789:−
3772:φ
3769:
3760:−
3757:φ
3754:
3723:−
3707:¯
3689:φ
3669:φ
3666:
3654:φ
3651:
3540:see above
3404:θ
3401:
3389:θ
3386:
3374:θ
3371:
3359:θ
3356:
3339:θ
3306:, giving
3170:θ
3156:θ
3153:
3141:θ
3138:
3132:−
3093:θ
3090:
3078:θ
3075:
3000:θ
2997:
2985:θ
2982:
2872:
2857:
2836:⋯
2808:−
2758:−
2736:⋯
2733:−
2683:−
2633:−
2612:⋯
2554:−
2529:−
2444:−
2419:−
2394:⋯
2074:⋮
2066:⋮
2058:⋮
2050:⋮
2037:−
2009:−
1929:−
1901:−
1771:θ
1768:
1756:θ
1753:
1742:θ
1670:θ
1667:
1655:θ
1652:
1639:θ
1633:−
1622:−
1614:θ
1611:
1605:−
1602:θ
1599:
1583:θ
1577:−
1563:θ
1537:for real
1520:θ
1517:
1505:θ
1502:
1489:θ
1483:−
1468:θ
1456:θ
1453:
1441:θ
1438:
1423:θ
1316:∞
1313:→
1210:∞
1195:∑
1188:⋯
822:
777:∫
714:−
621:θ
618:
575:θ
572:
549:θ
505:
490:
481:
444:−
306:
291:
123:Defining
8335:Analysis
8308:Category
8046:Archived
7987:Calculus
7441:is thus
7076:See also
6799:3-sphere
6611:, where
6466:morphism
6422:topology
4464:analysis
4292:for any
3968:= atan2(
3910:argument
3848:= |
3263:. Thus,
2934:for some
1556:′
348:are the
249:and the
31:Part of
8113:3791469
7868:(1748)
6971:addends
6967:= 1 + 0
6948:(where
6918:
6904:
6866:
6852:
6710:In the
6644:below:
6390:removed
6375:sources
4045:
4029:
3959:
3947:
3926:radians
3908:is the
3868:is the
3856:√
3544:complex
3516:(where
3498:radians
3488:is the
3476:in the
3297:(0) = 0
3290:(0) = 1
3258:
3246:
3237:
3225:
1711:, then
1709:(0) = 1
1380:means.
426:History
409:+ 1 = 0
336:is the
328:is the
251:complex
239:formula
234:, is a
8111:
8101:
8082:
7994:
7967:
7923:
7853:
7824:
7607:
7480:
7324:&
7233:&
7157:
7130:
6958:, one
6795:versor
6720:sphere
6584:kernel
6460:is a (
3944:= tan
3883:= arg
3816:where
3440:= cos
1693:Thus,
1384:Proofs
1368:Here,
936:unique
895:, and
461:) as:
353:cosine
340:, and
324:where
260:
159:People
109:growth
35:on the
7792:p. 28
7769:See:
6836:0 + 2
6545:as a
3930:up to
3889:atan2
3836:= Im
3824:= Re
3490:angle
3468:is a
3201:(cos
1720:) = 1
1001:of a
394:When
376:is a
113:decay
8099:ISBN
8080:ISBN
8054:2013
8024:2024
7992:ISBN
7965:ISBN
7921:ISBN
7902:1702
7851:ISBN
7822:ISBN
7437:and
7155:ISBN
7128:ISBN
6960:turn
6809:The
6373:any
6371:cite
6331:cos
6313:cos
5184:sinh
5105:cosh
5066:sinh
4944:cosh
4466:and
4165:and
4023:tan
4005:and
3989:and
3529:cos
3527:and
3522:sin
3448:sin
3313:and
3292:and
3242:and
3215:for
3209:sin
3024:and
2939:and
2907:sin
2905:and
2900:cos
2898:for
1803:for
1048:and
416:= −1
357:sine
355:and
344:and
111:and
7872:of
7726:sin
7704:cos
7593:sin
7571:cos
7471:sin
7449:cos
7066:= 1
7020:sin
7005:cos
6953:= 2
6920:+ 2
6888:+ 2
6868:+ 2
6772:sin
6757:cos
6742:exp
6722:of
6714:of
6687:In
6666:In
6549:of
6468:of
6384:by
6263:cos
6248:cos
6209:cos
6154:cos
6055:cos
5829:cos
5769:cos
5745:cos
5305:cos
5296:cos
5250:sin
5168:cos
4960:sin
4887:cos
4836:sin
4821:cos
4800:sin
4776:cos
4737:sin
4722:cos
4580:sin
4486:cos
4297:≠ 0
4011:, −
3983:≤ 0
3912:of
3876:and
3872:of
3766:sin
3751:cos
3663:sin
3648:cos
3480:as
3398:sin
3383:cos
3368:sin
3353:cos
3311:= 1
3304:= 1
3271:is
3261:= 1
3240:= 0
3150:cos
3135:sin
3087:sin
3072:cos
3034:is
2994:sin
2979:cos
2869:sin
2854:cos
1808:= 2
1765:sin
1750:cos
1664:sin
1649:cos
1608:sin
1596:cos
1514:sin
1499:cos
1450:sin
1435:cos
1306:lim
941:of
615:sin
569:cos
502:sin
487:cos
411:or
362:cis
346:sin
342:cos
303:sin
288:cos
241:in
8326::
8109:MR
8107:.
8044:.
8015:.
7900:.
7846:,
7806:^
7778:29
7776:,
7694:CE
7559:ln
7531:•
7529:CE
7494:CE
7439:CE
7393:XE
7381:CX
7373:CE
7361:CE
7326:CE
7280:CE
7276:XE
7272:CX
7268:CE
7264:."
7235:CE
7189:CE
7185:XE
7181:CX
7177:CE
7072:.
6943:=
6922:πn
6908:3π
6896:−1
6890:πn
6870:πn
6838:πn
6691:,
6684:.
6464:)
6315:nx
6109:Re
5958:Re
5888:Re
5848:Re
4872:iy
4870:=
4596:Im
4502:Re
4445:.
4319:ln
4307:ln
4255:ln
4215:ln
4079:ln
4048:.
4027:=
4007:(−
3999:,
3972:,
3895:,
3887:=
3861:+
3592:iy
3590:−
3586:=
3576:iy
3574:+
3570:=
3550:.
3500:.
3444:+
3318:=
3276:+
3255:dx
3249:dθ
3234:dx
3228:dr
3205:+
3037:ie
2947:,
2920:.
2912:.
1822::
1810:.
1273:.
1071:1.
883:.
819:ln
648:.
633:).
534:.
531:πi
478:ln
422:.
399:=
332:,
8155:e
8148:t
8141:v
8115:.
8088:.
8056:.
8026:.
8000:.
7973:.
7929:.
7830:.
7794:.
7748:1
7739:e
7735:=
7721:1
7713:+
7678:1
7651:1
7623:)
7620:E
7617:C
7614:(
7611:=
7603:)
7588:1
7580:+
7567:(
7556:E
7553:C
7548:1
7533:θ
7525:θ
7509:1
7466:1
7458:+
7423:1
7415:C
7412:X
7409:+
7406:X
7403:E
7389:X
7385:E
7377:C
7369:E
7365:θ
7357:E
7341:1
7310:1
7302:C
7299:X
7296:+
7293:X
7290:E
7250:1
7219:1
7211:C
7208:X
7205:+
7202:X
7199:E
7163:.
7136:.
7064:e
7045:0
7042:+
7039:1
7036:=
7017:i
7014:+
7002:=
6990:i
6986:e
6965:e
6955:π
6951:τ
6945:τ
6941:x
6930:i
6928:−
6915:2
6912:/
6886:π
6877:i
6863:2
6860:/
6856:π
6844:1
6828:e
6821:x
6781:,
6778:x
6769:r
6766:+
6763:x
6754:=
6751:r
6748:x
6732:x
6728:r
6673:e
6625:2
6622:=
6598:Z
6564:1
6559:S
6532:R
6509:1
6504:S
6481:R
6446:t
6443:i
6439:e
6432:t
6411:)
6405:(
6400:)
6396:(
6392:.
6378:.
6333:x
6324:x
6320:n
6293:.
6290:]
6287:x
6284:)
6281:2
6275:n
6272:(
6269:[
6257:]
6254:x
6245:2
6242:[
6236:]
6233:x
6230:)
6227:1
6221:n
6218:(
6215:[
6206:=
6195:)
6189:x
6186:)
6183:2
6177:n
6174:(
6171:i
6167:e
6160:x
6151:2
6143:x
6140:)
6137:1
6131:n
6128:(
6125:i
6121:e
6116:(
6106:=
6094:)
6087:)
6080:x
6077:i
6070:e
6061:x
6052:2
6040:x
6037:i
6030:e
6026:+
6021:x
6018:i
6014:e
6004:(
5994:x
5991:)
5988:1
5982:n
5979:(
5976:i
5972:e
5966:(
5955:=
5944:)
5938:x
5935:i
5931:e
5922:x
5919:)
5916:1
5910:n
5907:(
5904:i
5900:e
5895:(
5885:=
5874:)
5869:x
5866:n
5863:i
5859:e
5855:(
5845:=
5838:x
5835:n
5795:.
5791:)
5787:)
5784:y
5778:x
5775:(
5766:+
5763:)
5760:y
5757:+
5754:x
5751:(
5741:(
5735:2
5732:1
5727:=
5715:)
5708:2
5702:)
5699:y
5693:x
5690:(
5687:i
5680:e
5676:+
5671:)
5668:y
5662:x
5659:(
5656:i
5652:e
5645:+
5640:2
5634:)
5631:y
5628:+
5625:x
5622:(
5619:i
5612:e
5608:+
5603:)
5600:y
5597:+
5594:x
5591:(
5588:i
5584:e
5575:(
5568:2
5565:1
5560:=
5548:2
5542:)
5539:y
5533:x
5527:(
5524:i
5520:e
5516:+
5511:)
5508:y
5505:+
5502:x
5496:(
5493:i
5489:e
5485:+
5480:)
5477:y
5471:x
5468:(
5465:i
5461:e
5457:+
5452:)
5449:y
5446:+
5443:x
5440:(
5437:i
5433:e
5421:2
5418:1
5413:=
5401:2
5395:y
5392:i
5385:e
5381:+
5376:y
5373:i
5369:e
5357:2
5351:x
5348:i
5341:e
5337:+
5332:x
5329:i
5325:e
5318:=
5311:y
5302:x
5259:.
5256:x
5247:i
5244:=
5239:2
5233:x
5230:i
5223:e
5214:x
5211:i
5207:e
5200:=
5193:x
5190:i
5177:,
5174:x
5165:=
5160:2
5154:x
5151:i
5144:e
5140:+
5135:x
5132:i
5128:e
5121:=
5114:x
5111:i
5075:.
5072:y
5063:i
5060:=
5057:i
5052:2
5046:y
5039:e
5030:y
5026:e
5019:=
5013:i
5010:2
5003:y
4999:e
4990:y
4983:e
4976:=
4969:y
4966:i
4953:,
4950:y
4941:=
4936:2
4930:y
4926:e
4922:+
4917:y
4910:e
4903:=
4896:y
4893:i
4868:x
4863:x
4842:x
4833:i
4827:x
4818:=
4815:)
4812:x
4806:(
4797:i
4794:+
4791:)
4788:x
4782:(
4773:=
4764:x
4761:i
4754:e
4746:,
4743:x
4734:i
4731:+
4728:x
4719:=
4710:x
4707:i
4703:e
4672:.
4666:i
4663:2
4656:x
4653:i
4646:e
4637:x
4634:i
4630:e
4623:=
4619:)
4614:x
4611:i
4607:e
4603:(
4593:=
4586:x
4573:,
4568:2
4562:x
4559:i
4552:e
4548:+
4543:x
4540:i
4536:e
4529:=
4525:)
4520:x
4517:i
4513:e
4509:(
4499:=
4492:x
4435:k
4421:,
4416:k
4413:a
4409:e
4405:=
4400:k
4395:)
4390:a
4386:e
4382:(
4367:φ
4345:,
4339:i
4336:+
4332:|
4329:z
4326:|
4316:=
4313:z
4295:z
4275:i
4272:+
4268:|
4265:z
4262:|
4251:e
4247:=
4239:i
4235:e
4228:|
4225:z
4222:|
4211:e
4207:=
4199:i
4195:e
4190:|
4187:z
4184:|
4180:=
4177:z
4167:b
4163:a
4149:,
4144:b
4141:+
4138:a
4134:e
4130:=
4125:b
4121:e
4115:a
4111:e
4090:,
4085:a
4075:e
4071:=
4068:a
4041:x
4037:/
4033:y
4025:φ
4019:π
4015:)
4013:y
4009:x
4003:)
4001:y
3997:x
3995:(
3991:y
3987:x
3981:x
3976:)
3974:x
3970:y
3966:φ
3956:x
3953:/
3950:y
3942:φ
3936:π
3934:2
3922:z
3918:x
3914:z
3906:φ
3901:.
3899:)
3897:x
3893:y
3891:(
3885:z
3881:φ
3874:z
3863:y
3859:x
3851:z
3846:r
3838:z
3834:y
3826:z
3822:x
3800:,
3792:i
3785:e
3781:r
3778:=
3775:)
3763:i
3748:(
3744:|
3740:z
3736:|
3732:=
3729:y
3726:i
3720:x
3717:=
3704:z
3694:,
3686:i
3682:e
3678:r
3675:=
3672:)
3660:i
3657:+
3645:(
3641:|
3637:z
3633:|
3629:=
3626:y
3623:i
3620:+
3617:x
3614:=
3607:z
3588:x
3583:z
3572:x
3568:z
3548:x
3538:(
3536:x
3531:x
3524:x
3518:z
3513:e
3486:φ
3482:φ
3465:e
3450:φ
3446:i
3442:φ
3438:e
3407:.
3395:i
3392:+
3380:=
3377:)
3365:i
3362:+
3350:(
3347:1
3344:=
3336:i
3332:e
3320:x
3316:θ
3309:r
3302:e
3295:θ
3288:r
3283:C
3278:C
3274:x
3269:θ
3265:r
3252:/
3231:/
3218:e
3213:)
3211:θ
3207:i
3203:θ
3199:r
3184:.
3178:x
3175:d
3167:d
3160:)
3147:i
3144:+
3128:(
3124:r
3121:+
3115:x
3112:d
3107:r
3104:d
3097:)
3084:i
3081:+
3068:(
3064:=
3059:x
3056:i
3052:e
3048:i
3031:e
3026:θ
3022:r
3008:.
3004:)
2991:i
2988:+
2975:(
2971:r
2968:=
2963:x
2960:i
2956:e
2945:x
2941:θ
2937:r
2909:x
2902:x
2878:,
2875:x
2866:i
2863:+
2860:x
2851:=
2840:)
2833:+
2827:!
2824:7
2818:7
2814:x
2802:!
2799:5
2793:5
2789:x
2783:+
2777:!
2774:3
2768:3
2764:x
2755:x
2751:(
2747:i
2744:+
2740:)
2727:!
2724:8
2718:8
2714:x
2708:+
2702:!
2699:6
2693:6
2689:x
2677:!
2674:4
2668:4
2664:x
2658:+
2652:!
2649:2
2643:2
2639:x
2630:1
2626:(
2622:=
2609:+
2603:!
2600:8
2594:8
2590:x
2584:+
2578:!
2575:7
2568:7
2564:x
2560:i
2548:!
2545:6
2539:6
2535:x
2523:!
2520:5
2513:5
2509:x
2505:i
2499:+
2493:!
2490:4
2484:4
2480:x
2474:+
2468:!
2465:3
2458:3
2454:x
2450:i
2438:!
2435:2
2429:2
2425:x
2416:x
2413:i
2410:+
2407:1
2404:=
2391:+
2385:!
2382:8
2375:8
2371:)
2367:x
2364:i
2361:(
2355:+
2349:!
2346:7
2339:7
2335:)
2331:x
2328:i
2325:(
2319:+
2313:!
2310:6
2303:6
2299:)
2295:x
2292:i
2289:(
2283:+
2277:!
2274:5
2267:5
2263:)
2259:x
2256:i
2253:(
2247:+
2241:!
2238:4
2231:4
2227:)
2223:x
2220:i
2217:(
2211:+
2205:!
2202:3
2195:3
2191:)
2187:x
2184:i
2181:(
2175:+
2169:!
2166:2
2159:2
2155:)
2151:x
2148:i
2145:(
2139:+
2136:x
2133:i
2130:+
2127:1
2124:=
2115:x
2112:i
2108:e
2094:x
2040:i
2034:=
2025:7
2021:i
2015:,
2012:1
2006:=
1997:6
1993:i
1987:,
1984:i
1981:=
1972:5
1968:i
1962:,
1959:1
1956:=
1947:4
1943:i
1935:,
1932:i
1926:=
1917:3
1913:i
1907:,
1904:1
1898:=
1889:2
1885:i
1879:,
1876:i
1873:=
1864:1
1860:i
1854:,
1851:1
1848:=
1839:0
1835:i
1820:i
1806:t
1800:e
1774:.
1762:i
1759:+
1747:=
1739:i
1735:e
1724:θ
1718:θ
1716:(
1714:f
1707:f
1702:)
1700:θ
1698:(
1696:f
1681:0
1678:=
1674:)
1661:i
1658:+
1645:(
1636:i
1629:e
1625:i
1618:)
1593:i
1589:(
1580:i
1573:e
1569:=
1566:)
1560:(
1553:f
1539:θ
1524:)
1511:i
1508:+
1495:(
1486:i
1479:e
1475:=
1465:i
1461:e
1447:i
1444:+
1429:=
1426:)
1420:(
1417:f
1408:)
1406:θ
1404:(
1402:f
1378:n
1370:n
1354:.
1349:n
1344:)
1338:n
1335:z
1330:+
1327:1
1323:(
1310:n
1302:=
1297:z
1293:e
1283:z
1271:z
1266:e
1237:.
1231:!
1228:n
1222:n
1218:z
1205:0
1202:=
1199:n
1191:=
1185:+
1179:!
1176:3
1170:3
1166:z
1160:+
1154:!
1151:2
1145:2
1141:z
1135:+
1129:!
1126:1
1122:z
1117:+
1114:1
1111:=
1106:z
1102:e
1092:z
1068:=
1065:)
1062:0
1059:(
1056:f
1036:f
1033:=
1027:z
1024:d
1019:f
1016:d
983:z
979:e
975:=
972:)
969:z
966:(
963:f
944:e
931:x
927:z
923:z
918:e
909:x
904:e
849:,
846:C
843:+
840:)
837:x
834:a
831:+
828:1
825:(
814:a
811:1
806:=
800:x
797:a
794:+
791:1
786:x
783:d
755:.
751:)
744:x
741:i
738:+
735:1
731:1
726:+
720:x
717:i
711:1
707:1
701:(
695:2
692:1
687:=
679:2
675:x
671:+
668:1
664:1
595:i
529:2
514:.
511:)
508:x
499:i
496:+
493:x
484:(
475:=
472:x
469:i
447:1
414:e
407:e
401:π
397:x
374:x
370:i
365:x
334:i
326:e
312:,
309:x
300:i
297:+
294:x
285:=
280:x
277:i
273:e
262:x
216:e
209:t
202:v
145:e
136:e
125:e
43:e
23:.
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