In this chapter, first, the Mittag-Leffler functions, \(E_{\alpha }(z)\) , and their various generalizations: \(E_{\alpha , \beta }(z)\) by Wiman, \(E_{\alpha , \beta }^{\gamma }(z)\) by Prabhakar, and \(E_{\alpha , \beta }^{\gamma , q}(z)\) by Shukla-Prajapati are defined. Then, the generalized Bessel functions, \(\phi (\rho , \beta ; z)\) , by Wright are defined. Finally, the Pseudo-Exponential functions, which look very similar to the above functions, are defined. Using the theorems proved in Chap. 4, the integral representations of the pseudo-exponential functions are derived. The derivatives and integrals of the pseudo-exponential functions are also found and are shown to be very similar to the elementary exponential functions. The problems listed in this chapter show the relations between the pseudo-exponential functions and the elementary exponential functions. In particular, the pseudo-exponential functions are shown to be the special integrals of the elementary exponential functions. Many special integrals are shown to be simplified using the pseudo-exponential functions.