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Let $x\in \mathbb{R}^n$ and $h(x)=\left\{\begin{array}{cc} 1 & ,|x|<1\\ 0 & ,|x|\geq 1 \end{array}\right.$

I want to find the Fourier transform , $\hat{h}(\xi).$

Here is how I proceed $$\hat{h}(\xi)=\frac{1}{\left(2\pi\right)^{\frac{n}{2}}}\int_{\mathbb{R}^n}h(x)e^{-i\xi\cdot x}dx=\frac{1}{\left(2\pi\right)^{\frac{n}{2}}}\int_{|x|<1}e^{-i\xi\cdot x}dx$$ where $\xi\cdot x=\sum_{k=1}^n\xi_kx_k$ and $dx=dx_1\ldots dx_n$

I need some hints on how to evaluate the integral $$\frac{1}{\left(2\pi\right)^{\frac{n}{2}}}\int_{|x|<1}e^{-i\xi\cdot x}dx$$

Thomas Produit
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  • Don't you have here $e^{-i\xi\cdot x}\mathrm dx=\prod_{k=1}^ne^{-i\xi_k x_k}\mathrm dx_k$? – Caran-d'Ache Feb 08 '14 at 09:45
  • @Caran-d'Ache. This is right but not adapted to the prevailing symmetry. Hyperspherical coordinates should do the trick. See http://en.wikipedia.org/wiki/Hyperspherical_coordinates#Hyperspherical_coordinates – Urgje Feb 08 '14 at 11:33
  • @ Urgje.My problem is to find the trick that makes the integral simple. I looked at en.wikipedia.org/wiki/ , but still I can't fix the problem. Please help me in this regard. – Math Feb 10 '14 at 07:34

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