Image of Complex Exponential Function

Theorem

The image of the complex exponential function is $\C \setminus \set 0$.

Let $z \in \C \setminus \set 0$.

Let $r = \cmod z$ denote the modulus of $z$.

Let $\theta = \map \arg z$ denote the argument of $z$.

Then $r > 0$.

Let $\ln$ denote the real natural logarithm.

Let $e$ denote the real exponential function.

Then:

$\ds \map \exp {\ln r + i \theta}$

$=$

$\ds e^{\ln r} \paren {\cos \theta + i \sin \theta}$

Definition of Complex Exponential Function

$\ds $

$=$

$\ds r \paren {\cos \theta + i \sin \theta}$

$\ds $

$=$

$\ds z$

Definition of Polar Form of Complex Number

Hence, $z \in \Img \exp$.

Suppose instead that $z = 0$.

Let $z_0 = r_0 \paren {\cos \theta_0 + i \sin \theta_0} \in \C$.

From Exponential Tends to Zero and Infinity, it follows that $e^{r_0} \ne 0$.

As $\cmod {\cos \theta_0 + i \sin \theta_0} = 1$, it follows that:

$\cos \theta_0 + i \sin \theta_0 \ne 0$

Then this equation has no solutions:

$0 = \exp z_0 = e^{r_0} \paren {\cos \theta_0 + i \sin \theta_0}$

Hence:

$\Img \exp = \C \setminus \set 0$

$\blacksquare$

2001: Christian Berg: Kompleks funktionsteori: $\S 1.5$
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): complex exponential