Why isn't Euler's formula multivalued?

Question

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So it seems that all complex exponential functions are multivalued except for ones with base $e$. Why? Shouldn't all exponentials be multivalued?

I'm not clear on your question: the text you cite is not principally about complex exponential functions like $a^z$ but rather complex power functions like $z^\alpha$. — Semiclassical, Oct 19 '14 at 23:15
@Semiclassical Still, fix $z$ and evaluate $w^z$ at $w = e$. — Daniel Fischer, Oct 19 '14 at 23:16

score 3 · Answer 1 · answered Oct 19 '14 at 23:19

So it seems that all complex exponential functions are multivalued except for ones with base $e$. Why? Shouldn't all exponentials be multivalued?

Except when the exponent is an integer. Yes, but there is a very strong convention, that $e^z$ always means

$$\exp(z) := \sum_{n=0}^\infty \frac{z^n}{n!}$$

and no other value of $w\mapsto w^z$ evaluated at $w = e$. That convention should however be explicitly introduced lest it lead to confusion if it is tacitly used.

Why? Confusion is good. Chaos is even better! – Lucian Oct 20 '14 at 05:21 — Lucian, Oct 20 '14 at 05:21

Why isn't Euler's formula multivalued?

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