Mittag-Leffler functions as impropers integrals of trigonometric functions

• Grant number: 16/04888-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: June 01, 2016
• End date: November 30, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Eliana Contharteze Grigoletto
• Grantee: Luiz Carlos Silva Santana
• Host Institution: Faculdade de Ciências Agronômicas (FCA). Universidade Estadual Paulista (UNESP). Campus de Botucatu. Botucatu , SP, Brazil

Abstract

Expressions for improper integrals can be found in the book Table of Integrals, Series, and Products of authors Gradshteyn and Ryzhik's. New expressions for improper integrals of trigonometric functions in terms of functions involving the real Mittag-Leffler function, which represent a generalization of the exponential function and is used in fractional calculus, will be obtained in this project through the inverse Laplace transform of the methodology without using an integration contour. It is also intended relates to the convergents improper integrals with convergents series and its possible applications. Still using this method of the inversion of the Laplace transform without contour integration, it is intended to explain the solution of a particular telegraph equation as an alternative method to analytically solve a differential equation.