Definition:Power (Algebra)/Real Number/Complex

Definition

Let $x \in \R$ be a real number such that $x > 0$.

Let $r \in \C$ be any complex number.

Then we define $x^r$ as:

$x^r := \map \exp {r \ln x}$

where $\exp$ denotes the complex exponential function.

When $x = e$ this reduces to the definition of the complex exponential function.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $3$