This thesis presents a method for computing symbolic solutions of a certain class of improper integrals related to convolutions of Mellin transforms. Important integrals that fall into this category are integral transforms such as the Fourier, Laplace, and Hankel transforms. The method originated in a presentation by Salvy, However, many of the details of the method were absent. We present the method of Salvy in full which computes a linear homogeneous differentail equation which is satisfied by the integral in question. A theory of contour integrals is introduced that is related to the contour definition of Meijer G functions. This theory is used to prove the correctness of the method of Salvy and also gives a way to compute regions of validity for the solutions computed. We then extend the method to compute symbolic solutions of the integral along with where the solutions are valid.

This thesis presents a method for computing symbolic solutions of a certain class of improper integrals related to convolutions of Mellin transforms. Important integrals that fall into this category are integral transforms such as the Fourier, Laplace, and Hankel transforms. The method originated in a presentation by Salvy, However, many of the details of the method were absent. We present the method of Salvy in full which computes a linear homogeneous differentail equation which is satisfied by the integral in question. A theory of contour integrals is introduced that is related to the contour definition of Meijer G functions. This theory is used to prove the correctness of the method of Salvy and also gives a way to compute regions of validity for the solutions computed. We then extend the method to compute symbolic solutions of the integral along with where the solutions are valid.