Mellin Transforms in N Complex Planes
In Chaps. 4 and 7 Mellin transform techniques were used to solve problems of wave propagation through turbulence in which there was zero or one parameter. Here, that technique is generalized to allow one to solve problems with any number of parameters. Analytical solutions that are easily evaluated are obtained for problems that have previously been considered analytically intractable and for which only numerical results are available. The method to obtain the solution in integral form by the insertion of the transverse spatial filter functions into the standard formula for variances is exactly the same as that discussed in Chap. 2. This process produces a three-dimensional integral over the transverse spatial-transform coordinates and the propagation path. The integration over angle in transform space can usually be performed analytically. The integration over the magnitude of the spatial transform coordinate in which there are two or more parameters in the integrand is addressed in this chapter. I show that it can be evaluated to give a series solution. The remaining integration over the propagation path can be performed term by term in most cases. For some cases the infinite series terms after the axial integration are infinite, in which cases one must evaluate the integration along the propagation direction first.
KeywordsPower Series Complex Plane Complex Variable Gamma Function Asymptotic Solution
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