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

          Today´s post is a very special one for me. It took me a very long time to proof these Bessel functions integral representations and be able to present them the way I show here . To begin, we start proving Hankel´s contour integral representation for the Gamma Function which serves as the basis to derive the other integral representations for the Bessel function. Then, we  proof some integral representations for the Modified Bessel function of the first and second kind. Finally, we conclude the post computing Basset´s integral that shows up in probability as the characteristic function of the Student´s T distribution with the help of the integrals proved in the previous sections.   Hankel´s contour integral for The Gamma function The Gamma function has the following integral representation : (1) Proof: Let (2) And   be the following contour For the first integral on the line below the negative Real axis we have Then ...