◀ ▲ ▶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 $k\in\mathbb Z.$

Table of Contents

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Proofs: 1 2

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983