Expansion of 1/|x-x'| into Legendre Polynomials

[deuteron]

TL;DR Summary

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we know that we can expand the following function in Legendre polynomials in the following way

in the script given yo us by my professor, ##\frac 1 {|\vec x -\vec x'|}## is expanded using geometric series in the following way:

However, I don't understand how ##\frac 1 {|\vec x -\vec x'|}## is equal to both the above, and the below:

 

[kuruman]

Do you understand the meaning of ##\Theta(x-x')##?
It's not equal to both. It's equal to one or the other depending on which of ##x## and ##x'## is larger. The bottom expression summarizes in one line the two "für" cases above it.