MY MATEX NOTES

      Today we will proof the beautiful integral involving the modified Bessel function of the second kind, namely: We substitue the integral representation of      proved here , then Now, let      The last integral is a representation of the Beta function proved below in the appendix, then And we finally obtain (1) Special case, let in (1) we obtain (2) If we change then make the change of variable we get an integral representation for in terms of Bessel functions: (3) Appendix Proof: Recall the integral representation for the Beta function Then, let in the second integral

The goal is to compute the following Integral Lets start by considering the pair of Mellin Transforms proved here : (1) (2) set       in (1) and (2) (3) (4) Now subtract (3) from(4)

               Today I want to compute two pairs of integrals using two different techniques. Firts pair is the Mellin transform of sine and cosine using contour integration. The second one is a generalization of the first pair and will be evaluated through a mix of real and complex methods. In the end we will see how we can recover the first result from the second. The Mellin transforms of   and   are given by the following integrals: We first consider the following integral and the equate Real and Imaginary parts to get the desired results for Consider the following integral in the complex plain. For . C is the contour below Since there is no singularity inside the contour, by Cauchy´s theorem For the integral around the big arc let we have Since as this integral vanishes Similarly, letting in thefourth integral The integral is finite and as this term vanishes Therefore, after taking the limi...