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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?

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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.

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  • $\begingroup$ So should the formula in my question be understood only as a shorthand for the formula that you wrote? $\endgroup$
    – Holden
    Jan 22, 2020 at 22:35
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    $\begingroup$ 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. $\endgroup$ Jan 22, 2020 at 22:39

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