◀ ▲ ▶Branches / Analysis / Corollary: More Insight to Euler's Identity
The Euler's identity $\exp(i\pi)+1=0$ is only a special case of a more general result:# Corollary: More Insight to Euler's Identity
(related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
For some special real numbers $x\in\mathbb R$ the following equations hold:
$$\exp(ix)=\pm 1.\quad\quad (*) $$
These special cases for $x$ are as follows:
 If and only if $x=k\pi$ for all odd integers $k\in\mathbb Z$, we can put a "$+$" sign in the equation $(*),$
 otherwise, we can put a "$$" sign in the equation $(*)$ if and only if $x=k\pi$ for all even integers $k\in\mathbb Z.$
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983