Remarks
A Special Spherical Integral
I recently had to deal with the following surface integral (in computing the Fourier transform of the surface measure of a sphere): where the surface of the -ball centered at with radius , and the usual Euclidean “dot” product; namely and are all fixed. Finding a “nice” closed form for the expression proceeds as follows. First, we translate so the integral is about the origin, by sending , Next, notice that is rotationally symmetric. Namely, if is a rotation matrix, then where the hermitian adjoint of ; rotation matrices are unitary, so . Changing variables by letting , then, the new domain of integration remains since a rotation, and since , the resulting Jacobian of the transformation is unity. Thus, we find indeed.
From this observation, then, we may assume without loss of generality that is a scalar of the usual basis vector, , namely say , with the Euclidean norm. With this, with . Finally, rewriting this exponential using Euler’s formula and writing the integral using -dim spherical coordinates, we find (identifying ) The iterated product term simplifies using a well-known Gamma-function identity and the right-most integral is identically zero, since the sin terms are symmetric about . Finally, the first integral term is, up to a constant depending only on , , and , equal to the Bessel function (see here, section 3.3). All together, and explicitly writing out this constant, we find from which we arrive finally at a relatively-nice formula Throughout, we tacitly assumed . Indeed, if , the “unit ball” is just the open interval and so the surface of such a ball is the set , so
[*A similar computation can be made for the same integral but now over the ball . I’ll omit the details but describe some necessary computations. If we call the surface integral above (over the sphere ), the analogous integral can be written as which eventually leads to having to compute the integral (up to some constants related to , and the dimension )* *which citing again this reference, equals .*]