Fourier Transform of Bessel Function of Order Zero

We first prove the formula for the Bessel Integral representation of the Bessel function of order zero. To do this, we use a recursive version of integration by parts for powers of sin(x). We then consider the average value of the function exp(ix sin(theta)) over the unit circle. The Taylor series of this function and the aforementioned recursion formula prove that the average value is actually the Bessel function of order zero. This allows us to use the inverse Fourier transform to find the Fourier transform of J_0(x). #mikethemathematician, #mikedabkowski, #profdabkowski