The equation (14) and the equation (15) can be rewritten to obtain: Alternatively, and, In general, the inversion of the Laplace transform of any of the g functions has to be done via contour integration in the complex s plane . In addition, extensive mathematical tables are also available with many worked out examples $^{12}$. This section shows only one of the g functions, g$_{1}$$_{a}$ , as an illustration of the use of these tables (the rest of functions g$_{1}$$_{b}$ , g$_{2}$$_{a}$ , g$_{2}$$_{b}$ are easy to obtain due to their similarity). The following expressions can be found 12 7 f ( s ) F ( t ) 1 √ s e − k √ s 1 √$_{πt}$ e − k 2 4 t g$_{1}$ = g$_{1}$$_{a}$ + g$_{1}$$_{b}$ g$_{1}$$_{a}$ = − 1 2 P e − P ( x $^{′}$− x ) (20) g$_{1}$$_{b}$ = 1 2 P $_{e}$− P ( x + x $^{′}$) (21) g$_{2}$ = g$_{2}$$_{a}$ + g$_{2}$$_{b}$ g$_{2}$$_{a}$ = − 1 2 P e − P ( x − x $^{′}$) (22) g$_{2}$$_{b}$ = 1 2 P $_{e}$− P ( x + x $^{$^{′}$)}$. (23) for x < x ′ g$_{1}$ = − 1 2 P e − Px ′ ( e Px − e − Px ) (18) for x > x ′ g$_{2}$ = − 1 2 P ( e Px ′ − $_{e}$− Px ′ ) $_{e}$− $^{Px}$. (19)