The transcendental method for finding the analytical closed-form solution to the dispersive equation in the Linear Transport Theory is studied in some detail. The mathematical genesis of general form of the analytical solution is presented. This solution is exact according to the new closed-form representation of the numbers with desired accuracy in the transcendental method theory. This approach implies the qualitative improvement of the more conventional method such as successive approximations. The numerical results, obtained for different magnitudes of the cross section parameterc, support the validity and basic principles of the transcendental method. The conditions for the existence of the exact solution are discussed.