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I'm studying some Fourier analysis and have for a while been trying to figure out how to apply the Fourier transform to a phase modulated signal.

More rigorously stated, what is $$ \mathcal{F}[\,\cos(vt + f_m(t))\,] $$ where $$ f_m(t) = A \sin(v_At) + B \cos(v_Bt) $$

After trying several approaches the closest I got was noting that (for example) $\cos(A\sin(v_At))$ obviously has the same periodicity as $\sin(v_At)$, which gives some intuitive information as to what the solution should look like but is far from an actual solution.

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