Timeline for Integral over $S^{n-1}$ [duplicate]
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Jul 2, 2022 at 19:02 | history | closed |
Francois Ziegler CommunityBot |
Duplicate of Fourier transform of the unit sphere | |
| Jul 2, 2022 at 18:49 | comment | added | Carlo Beenakker | Without loss of generality you may orient the vector $w$ along the $x_1$ axis, then $$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw=\frac{2 \pi ^{\frac{n-1}{2}}}{\Gamma \left(\frac{n-1}{2}\right)}\int_0^\pi e^{i\lambda\cos\theta} \sin^{n-2}\theta\, d\theta$$ $$\qquad\qquad=(2 \pi )^{n/2} \lambda^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(\lambda).$$ | |
| Jul 2, 2022 at 18:05 | comment | added | Giorgio Metafune | This is the Fourier transform of the of the surface measure at the point $\lambda x$ and can be expressed through Bessel functions. See Stein-Weiss, Introduction to Fourier Analysis, pag 154 | |
| Jul 2, 2022 at 17:38 | history | edited | zoran Vicovic | CC BY-SA 4.0 |
added 18 characters in body
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| S Jul 2, 2022 at 17:37 | review | First questions | |||
| Jul 2, 2022 at 19:04 | |||||
| S Jul 2, 2022 at 17:37 | history | asked | zoran Vicovic | CC BY-SA 4.0 |