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The exponential is the inverse biholomorphism from the principal strip to the slit plane

Statement

Let

$P : = {w \in C : - π < Im w < π}, S : = C ∖ {x \in R : x \leq 0} .$

Then the restriction

$\exp : P \to S$

is a biholomorphism, and its inverse is the principal logarithm.

Facts & Assumptions

Given: The principal strip $P$ and slit plane $S$ above.

[F1]

The principal logarithm is a biholomorphism $Log : S \to P$ (The principal logarithm is a biholomorphism from the slit plane to the principal strip).

[F2]

The complex exponential is entire (The complex exponential is entire and its complex derivative is itself).

[F3]

A map is biholomorphic when it is bijective, holomorphic, and has holomorphic inverse (Biholomorphic maps between complex domains).

Proof

technique · direct

1.1F1given

By [F1], every $z \in S$ satisfies $Log z \in P$ and $\exp (Log z) = z$ , so $\exp : P \to S$ is surjective and $Log$ is a two-sided inverse candidate.

2.1F1step 1.1algebra

If $w \in P$ , then [F1] applied to $z = \exp w \in S$ gives $Log (\exp w) = w$ , so $\exp$ is injective on $P$ .

3.1F1F2F3step 1.1step 2.1∎

The restriction of $\exp$ to $P$ is holomorphic by [F2], and its inverse is the holomorphic map $Log$ by [F1]. Therefore [F3] makes $\exp : P \to S$ biholomorphic.

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §4.2 (standard reference, not scraped)