Abstract— A branch of mathematics of utmost importance to scientists and engineers concerned with real mathematical calculations is addressed in this study. The reader will find a systematic treatment here of the fundamental theory of the most important specific functions, as well as applications of theory to specific physics and engineering problems.

This research provides an introduction to the well-known classical special functions that play a role in mathematical physics, especially in major problems of boundary value. This branch of mathematics has a respectable history with great names, including Gauss, Euler, Fourier, Legendre, Bessel, and Riemann. All of them spent a lot of time on this topic. A good portion of their work was inspired by physics and the resulting differential equations. These activities culminated in the standard work of Whittaker and Watson about 70 years ago, A Course of Modern Analysis, which has had a great influence and is still important. As well as in applied fields such as electric current, fluid dynamics, heat conduction, wave equation, and quantum mechanics, special functions have extensive applications for more details in pure mathematics see([5],[8],[18],[21],[23]).
2.Gamma Function [8]
The Gamma function usually denoted by ( ) is seen as a generalization of the factorial. It’s was Euler (1707-1783) , a Swiss mathematician, who first worked on the curve of the function in 1729. The name and the notation (1752-1833) in 1809. This function is also called Euler Gamma function or the Euler Ian Integral of the second kind. Gamma function is defined by: ( ) ? ( )
The variable may be complex. The integral is absolutely convergent for ( ) .