Exponential on Complex Plane is Group Homomorphism

Theorem

Let $\struct {\C, +}$ be the additive group of complex numbers.

Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.

Let $\exp: \struct {\C, +} \to \struct {\C_{\ne 0}, \times}$ be the mapping:

$x \mapsto \map \exp x$

where $\exp$ is the complex exponential function.

Then $\exp$ is a group homomorphism.

Proof

If $z \in \C$, then by the definition of the complex exponential function, $\exp$ is a mapping $\C \to \C_{\ne 0}$.

Let $z_1, z_2 \in \C$.

By Exponential of Sum:

$\map \exp {z_1 + z_2} = \map \exp {z_1} \, \map \exp {z_2}$

Therefore $\exp: \struct {\C, +} \to \struct {\C_{\ne 0}, \times}$ is a group homomorphism.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47$. Homomorphisms and their elementary properties: Illustration