Closed form for Fourier transform-like Integral on $S^{n}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T02:42:06Z http://mathoverflow.net/feeds/question/118816 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118816/closed-form-for-fourier-transform-like-integral-on-sn Closed form for Fourier transform-like Integral on $S^{n}$ Italo 2013-01-13T15:30:03Z 2013-01-14T12:45:33Z <p>Hello!</p> <p>It may be a stupid question, i'm trying to find a closed form for an integral similar to a Fourier transform on $S^{n}$ but i'm stuck... Let $\alpha>0$, the integral i can't solve is</p> <p><code>$$I(p,\alpha)=\int_{S^{n}}e^{i\alpha\left&lt;p,q\right&gt;}d\mu_{S^{n}}(q)$$</code></p> <p>where $p,q\in S^{n}$, $\left&lt; \cdot,\cdot \right>$ is the euclidean scalar product on $\mathbb{R}^{n+1}$ and $d\mu_{S^{n}}$ is the measure induced by euclidean measure on $\mathbb{R}^{n+1}$. so the question is: is there a closed form for $I(p,\alpha)$ only in terms of $p$ and $\alpha$? </p> <p>Thank you in advance!</p> http://mathoverflow.net/questions/118816/closed-form-for-fourier-transform-like-integral-on-sn/118821#118821 Answer by Robert Bryant for Closed form for Fourier transform-like Integral on $S^{n}$ Robert Bryant 2013-01-13T16:02:10Z 2013-01-14T12:45:33Z <p>The integral obviously doesn't depend on $p$, since it must be rotationally invariant, and, using polar coordinates centered on $p$, one obtains $$I(p,\alpha) = \mathrm{vol}(S^{n-1})\int_{-1}^1 e^{i\alpha t}(1{-}t^2)^{(n-2)/2}dt =\frac{2\ \pi^{n/2}}{\Gamma(n/2)}\int_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(n-2)/2}dt.$$ This latter integral can be evaluated by standard techniques. For odd values of $n$, it is expressed in Bessel functions, while, for even values of $n$ it is expressed in terms of elementary functions. Thus, for $n=2$, one has $$\int_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(2-2)/2}dt = \frac{2\sin\alpha}{\alpha},$$ while, for $n=3$, one has $$\int_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(3-2)/2}dt = \frac{\pi\ J_1(\alpha)}{\alpha},$$ where $J_1$ denotes the Bessel function of the first kind, and so on.</p>