So I've spent the last few hours going around in circles trying to solve this. I have this equation:
$e^{(\cos{t})-1}$
of which I am trying to obtain the Fourier Transform for. What I have tried is start from the following (from the definition of the FT)
$F({\omega}) = \int e^{(\cos{t})-1} e^{-i\omega t} dt = \frac{1}{e}\int e^{(\cos{t})} e^{-i\omega t} dt $
I tried
- using the identity $cos(At) = \frac{e^{iAt}+e^{-iAt}}{2}$
which results in an exponential of an exponential of complex numbers. I couldn't get any further from here and none of my FT tables give anything for this kind of form
Following a method similar to the derivation of the FT of the Gaussian function by using integration by parts (page 4 of this pdf http://code.ucsd.edu/zeger/45/Gaussian.pdf )
Following the method of the last page of that pdf and splitting the Fourier $e^{-i\omega t}$ term into separate cosine and sin function and then use the Taylor expansion of cos but this still leaves the $e^{(\cos{t})}$ term. Perhaps that cosine term could be also split into an expansion however that looks like it could be messy.
Integration by substitution for the top function but this led nowhere.
I have also attempted to put this into Mathematica but itdoes not solve it and simply returns the input. My suspicion is this has something to do with the infinite integral.
If anyone could offer some advice on how to proceed it would be greatly appreciated!
regards
Since $e^{\cos(t)}$ is periodic, what you want is a Fourier series, not Fourier transform.
$$ \int_0^{2\pi} e^{\cos(t)-1} \exp(i k t)\; dt = 2 \pi I_k(1)/e$$ where $I_k$ is a modified Bessel function of order $k$.