Definition:Exponential Function/Complex/Real Functions

Definition

Let $\exp: \C \to \C \setminus \set 0$ denote the (complex) exponential function.

The exponential function can be defined by the real exponential, sine and cosine functions:

$\exp z := e^x \paren {\cos y + i \sin y}$

where $z = x + i y$ with $x, y \in \R$.

Here, $e^x$ denotes the real exponential function, which must be defined first.

The complex number $\exp z$ is called the exponential of $z$.

Also see

• Equivalence of Definitions of Complex Exponential Function

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Euler's Formula
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $3$