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Difference between revisions of "Euler formulas"

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Formulas connecting the exponential and trigonometric functions:
 
Formulas connecting the exponential and trigonometric functions:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036460/e0364601.png" /></td> </tr></table>
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$$e^{iz}=\cos z+i\sin z,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036460/e0364602.png" /></td> </tr></table>
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$$\cos z=\frac{e^{iz}+e^{-iz}}{2},\quad\sin z=\frac{e^{iz}-e^{-iz}}{2i}.$$
  
These hold for all values of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036460/e0364603.png" />. In particular, for a real value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036460/e0364604.png" /> the Euler formulas become
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These hold for all values of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036460/e0364605.png" /></td> </tr></table>
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$$\cos x=\frac{e^{ix}+e^{-ix}}{2},\quad\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$
  
 
These formulas were published by L. Euler in [[#References|[1]]].
 
These formulas were published by L. Euler in [[#References|[1]]].

Latest revision as of 12:50, 10 August 2014

Formulas connecting the exponential and trigonometric functions:

$$e^{iz}=\cos z+i\sin z,$$

$$\cos z=\frac{e^{iz}+e^{-iz}}{2},\quad\sin z=\frac{e^{iz}-e^{-iz}}{2i}.$$

These hold for all values of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become

$$\cos x=\frac{e^{ix}+e^{-ix}}{2},\quad\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$

These formulas were published by L. Euler in [1].

References

[1] L. Euler, Miscellanea Berolinensia , 7 (1743) pp. 193–242
[2] L. Euler, "Einleitung in die Analysis des Unendlichen" , Springer (1983) (Translated from Latin)
[3] A.I. Markushevich, "A short course on the theory of analytic functions" , Moscow (1978) (In Russian)


Comments

References

[a1] K.R. Stromberg, "An introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Euler formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_formulas&oldid=32798
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article