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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? –  Davide Giraudo Aug 5 '12 at 12:40
    
it is from a physics book. it seems the author does not worry to much about the dirichlet conditions. –  Hamurabi Aug 5 '12 at 12:56
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And how is the Fourier transform defined? –  Davide Giraudo Aug 5 '12 at 13:10
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What's the domain of $f(x)$? –  Mhenni Benghorbal Aug 5 '12 at 13:19
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What is the name of the physics book you got it from? –  Fabian Aug 5 '12 at 13:21

1 Answer 1

up vote 1 down vote accepted

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? –  Hamurabi Aug 5 '12 at 14:34
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If $d$ is not an integer, it looks rather difficult... –  Fabian Aug 5 '12 at 14:46
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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, $\hat{f}(\omega) = \sum_{n_1} \cdots \sum_{n_d} a_{n_1} \cdots a_{n_d} \delta(\omega - \frac{b_{n_1}+ \cdots + b_{n_d}}{2})$. –  copper.hat Aug 5 '12 at 15:03
    
right. d is supposed to be an integer. i accidentally forgot to note it down. –  Hamurabi Aug 5 '12 at 15:33
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@Hamurabi:It has uses in electrical engineering. Read here –  Mhenni Benghorbal Aug 5 '12 at 19:03

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