# Characteristic function of the gamma distribution

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?

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.
• Essentially, unless the exponent is an integer, $a^b$ is (for $a \neq 0$) always just a shorthand for $\exp (b\log a)$. Which of the infinitely many logarithms of $a$ is used in that needs to be stated, or deduced via convention. Jan 22, 2020 at 22:39