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I am looking for a nice way to calculate the FT of the following function

$f(x)=\biggl(\sum_{n=1}^{c}~a_n~e^{-\frac{i}{2}~x~b_n}\biggr)^d$,

where $d,c>0$, $a_n$ and $b_n$ are real coefficients, strictly monotonously rising in $n$ and $x$ is the free variable and $c$ might go to $\infty$.

I used mathematica to calculate it, but without specification of $d$ and when $c\to\infty$, there is no way that the programme will do it.

Any helpful ideas? Thanks!!

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    Are you sure $f$ is integrable? And is $d$ an integer?2012-08-05
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    it is from a physics book. it seems the author does not worry to much about the dirichlet conditions.2012-08-05
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    And how is the Fourier transform defined?2012-08-05
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    $\int_{-\infty}^{\infty}~dx~f(x)~e^{ikx}$ is not given.2012-08-05
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    What's the domain of $f(x)$?2012-08-05
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    What is the name of the physics book you got it from?2012-08-05
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    its from a statistical mechanics book. i know well that usually one wouldnt compute it straightforwardly. there is a similar one with $f(x)=\prod_{n=1}^d \frac{e^{-ix/2}}{1-e^{-ix}}$. It is said that it is computed with the "saddlepoint method", which is then done. It refers to de Bruijn,asymptotic methods in analysis and morese, feshbach methods in theoretical physics. I cant apply these techniques to the given function.2012-08-05
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    the domain of x should be going from 0 till infinity.2012-08-05

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If $d$ is a positive integer, then you can use the multinomial theorem to expand your expression then take the Fourier transform with the appropriate condition on $\sum b_k $.

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    ahh. so this would give me some sort of a comb of delta-distributions, right?2012-08-05
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    If $d$ is not an integer, it looks rather difficult...2012-08-05
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    If $d$ is a positive integer, expand $f$ as $f(x) = \sum_{n_1} \cdots \sum_{n_d} a_{n_1} \cdots a_{n_d} e^{\frac{i}{2} x (b_{n_1}+ \cdots + b_{n_d})}$, and then, formally, $\hat{f}(\omega) = \sum_{n_1} \cdots \sum_{n_d} a_{n_1} \cdots a_{n_d} \delta(\omega - \frac{b_{n_1}+ \cdots + b_{n_d}}{2})$.2012-08-05
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    right. d is supposed to be an integer. i accidentally forgot to note it down.2012-08-05
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    so this is what i meant with the comb of dirac deltas. if d was $\infty$ the comb is quite dense. can i use this in any sense?2012-08-05
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    @Hamurabi:Didn't I gave you the idea and the steps to solve your problem in my answer?2012-08-05
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    @Mhenni. Sure. I got it. and its right. the multinomial theorem is a nice trick. i just wanted to know, if I can make something from the delta comb. thats all.2012-08-05
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    @Hamurabi:It has uses in electrical engineering. Read [here](http://en.wikipedia.org/wiki/Dirac_comb)2012-08-05