Characteristic function of the gamma distribution

Question

The characteristic function of the gamma distribution with shape $\alpha>0$ and rate $\beta>0$ is given by $$ \varphi(x)=\frac{1}{\left( 1-\frac{ix}{\beta} \right)^{\alpha}}, \quad x\in \mathbb{R}. $$ I am wondering why is raising the complex number in the denominator to the power $\alpha>0$ well-defined?

Answer 1

Unless $\alpha$ is an integer, it's not well-defined as such, but by convention one uses the principal branch of the logarithm when that's applicable and nothing else is explicitly stated. Since $$\operatorname{Re} \biggl(1 - \frac{ix}{\beta}\biggr) = 1$$ for all $x \in \mathbb{R}$, the principal branch is applicable, and we can unambiguously write $$\varphi(x) = \exp \biggl(-\alpha\operatorname{Log}\biggl(1 - \frac{ix}{\beta}\biggr)\biggr)\,,$$ where $\operatorname{Log}$ denotes the principal branch of the logarithm.

Since we need $\varphi(0) = 1$ for a characteristic function, the principal branch is the only admissible choice to define the power, thus in context the power is well-defined, although viewed in isolation it isn't.