I'm doing a research project over the summer, and need some help understanding how to construct an inverse Fourier transform (I have v. little prior experience with them).(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

I know the explicit form of ##q(x)##, where

$$ q(x) = \frac{M}{2 \pi} \int _{- \infty}^{\infty} dz e^{-iMxz} C_q(z)

$$

and want to find ##C_q(z)## using an inverse Fourier transform. As far as I can tell, there's no simple relationship between ##z## and ##x##. And the domain of ##x## is ##[0,1]##.

2. Relevant Equations

Explicit form of ##q(x)##: ##q(x) = x^{1/5}(1-x)^3##.

3. The attempt at a solution

I thought I would start with a substitution, since ##z## and ##M## are independent: ##\mu = Mz##. Therefore,

$$q(x) = \frac{1}{2 \pi} \int _{- \infty}^{\infty} d\mu e^{-ix\mu} \tilde{C}_{q}(\mu)

$$

And from this relation I use the inverse Fourier transform to get

$$\tilde{C}_{q}(\mu) = \int _{0}^{1} dx e^{ix\mu} q(x)

$$

$$ \Rightarrow \quad C_{q}(z) = \int _{0}^{1} dx e^{iMxz} q(x)

$$

Is this reasoning sound? Any help is appreciated.

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# Homework Help: Fourier transform between variables of different domains

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