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I do believe that the title got most of my question covered,

$$\sum_{n=0}^{\infty}r^n\cos(wn+\phi)z^{-n}=?$$

Any help is highly appreciated.

My work so far; I could use trigonometric identity, but specifically I was wondering if this problem could be solved by using Euler's formula.

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Write $$\cos(w n + \phi) = \frac{e^{i (wn + \phi)}}{2} + \frac{e^{-i(wn+\phi)}}{2}$$ and you get two geometric series to sum. Be careful about convergence.

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The given summation is the real part of

$$e^{i\phi}\sum_{n=0}^\infty(re^{iw}/z)^n$$ which is s geometric series

Use https://en.m.wikipedia.org/wiki/Geometric_series#Proof_of_convergence

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