# Method for expanding an expression with function composition

Sorry for the bad title; I couldn't think of a good way to phrase it. Someone more knowledgeable could edit it if that helps. I'm an engineer, not a mathematician, so my level of rigor is somewhat lacking.

This question is motivated by my answer to this question at the Signal Processing Stack Exchange site. I currently have a poor handwaving explanation for how to express one function that is defined as the composition of two functions solely in terms of the independent variable used in the most inner layer of the function composition. More precisely:

$$Y_D(z) = X\left(z^{1/M}\right) = f(g(z))$$

where

$$f(\alpha) = X(\alpha)$$

$$g(\beta) = \beta^{1/M}$$

(with $z, \alpha, \beta \in \mathbb{C}$)

Specifically, I would like to arrive at an expression for $Y_D(z)$ that is solely in terms of the intermediate function $X(z)$. The question references a paper that indicates that it can be expressed as:

$$Y_D(z)=\frac{1}{M}\sum_{k=0}^{M-1}X(z^{1/M} e^{j2\pi k/M})$$

Which seems to make sense to me, but I'm not sure how to rigorously arrive at that result. In the past, I recall using a method to derive expressions for the pdf of functions of a random variables. That method involved something like breaking the function applied to the random variable into sections that are one-to-one and treating them separately. The individual results were summed together, with the absolute value of the derivative of the mapping function used as a scaling factor in there somewhere.

It seems to me that a similar method could be applied here; I understand that for any particular value $z$ that we would want to evaluate $Y_D(z)$ at, there are $M$ distinct values $z^{1/M}$ in the domain of $X(z^{1/M})$ that would map to it, hence the sum, with the $M$ roots of unity included to catch all of the $M$-th roots of $z$. I'm just not sure how to cleanly get there.

Does this seem to ring a bell to anyone? Is there a particular name for the method that would be applied to this sort of problem?

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