What is the values of the following integral:
$$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw.$$ where $\lambda\in\Bbb R, i^2=-1,x\in\Bbb R^n;<,>$ the inner product scalar on $\Bbb R^n$ and $S^{n-1}$ the unit sphere of $\Bbb R^n$.
$\begingroup$
$\endgroup$
2
-
2$\begingroup$ 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 $\endgroup$– Giorgio MetafuneCommented Jul 2, 2022 at 18:05
-
2$\begingroup$ Without loss of generality you may orient the vector $w$ along the $x_1$ axis, then $$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw=\frac{2 \pi ^{\frac{n-1}{2}}}{\Gamma \left(\frac{n-1}{2}\right)}\int_0^\pi e^{i\lambda\cos\theta} \sin^{n-2}\theta\, d\theta$$ $$\qquad\qquad=(2 \pi )^{n/2} \lambda^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(\lambda).$$ $\endgroup$– Carlo BeenakkerCommented Jul 2, 2022 at 18:49
Add a comment
|