I'm working through equations in "Nonsteady Flow To a Well in an Infinite Anisotropic Aquifer" (Papadopulos, 1965). I've have gotten stuck trying to figure out how the inverse transform given below is derived. I've looked through the transform tables in Churchill (1958) and Erdelyi (Vol1, 1954) but haven't made much headway. Is there anyone who can help me figure it out?
Eqn(10), the transform solution for the presented groundwater flow equation, is given as:
$$z = {\frac{Q}{2\pi p}} {\frac{1}{T_{xx}\alpha^2+2T_{xy}\alpha\beta+T_{yy}\beta^2+Sp}}$$
where $\beta$ is the parameter of transformation and the remaining parameters can be considered as constants for transform purposes.
Eqn(11) is stated to be the inverse transform of the above with respect to y:
$$w={\frac{Q}{2 \sqrt{2\pi}}}{\frac{exp\{(iy\alpha T_{xy}-\mid y\mid [\ (T_{xx}T_{yy}-T_{xy}^2) \alpha^2+ST_{yy}p]\ ^{1/2})/T_{yy}\}}{p[\ (T_{xx}T_{yy}-T_{xy}^2) \alpha^2+ST_{yy}p ]\ ^{1/2}}}$$
The answer is that the inverse was derived using the Residue Method. The author of the paper was kind enough to pass on his notes to me; I'll include an update here once I've had a chance to sit down and work through them.