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In my physics problem, I encountered a solution has the form like Cos[Sin[t]], and I need to do the Fourier transform to this solution. Is there a way to do the Fourier transform analytically to Cos[Sin[t]] on a certain range, say from t=0 to t=2$\pi$? Or is there some way to manipulate Cos[Sin[x]] and get a approximate expresion which can be transformed analytically? The problem maybe rephrased as to do the integral analytically or approximately:

$$ \int_0^{2\pi}\cos{(\sin{(t)}})~e^{i\omega t}dt $$

Thanks.

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