Euler formulas

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)

References

[a1]	K.R. Stromberg, "An introduction to classical real analysis" , Wadsworth (1981)