On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms

In this paper we introduce decomposition functions C_Γ(α,x;ω), S_Γ(α,x;ω), C_γ(α,x;ω) and S_γ(α,x;ω) of the generalized gamma functions. These functions are found useful in the analytic study of the temperature distribution of a semi-infinite solid with periodic boundary conditions and to the theory of Fourier transforms. Several new identities involving the Fourier transforms are investigated and some of the classical ones are recovered as special cases. For numerical and scientific computations, tabular and graphical representations of the functions C_Γ(α,x;ω) and S_Γ(α,x;ω) are also given.