Euler formulas

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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].


[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)



[a1] K.R. Stromberg, "An introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Euler formulas. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article