Different Alternate Representations of Functions

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Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions?

For example, I know of two such ways

1) Taylor Series Expansion

2) Fourier Transform

Any suggestions for other such methods would be highly appreciated. Please note, I am looking to solve the problem from the linked question below and hence looking to see if any such alternate representation of functions could be helpful towards the simplification of the below.

Origin of Question:

Simplifying Multiple Integral for Compound Probability Density Function