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Chapter 8: Mellin Transforms with a Complex Parameter
When one analyzes beam-wave propagation through turbulence, the propagation parameter Î³ is complex. This causes complications when one evaluates the steepest-descent contribution to an asymptotic solution. In some problems, the spatial filter functions contain a decaying exponential times a function that grows exponentially when Î³ is complex. Consequently, the Mellin convolution relation must be generalized to allow for convergent integrals that contain integrands that are the product of functions that exhibit exponential growth and exponential decay. In addition, methods developed in Chap. 5 must be extended to include asymptotic results for a non-positive parameter. This requires an extension of saddle-point results to complex-parameter cases. These extensions are discussed in this chapter.
The theory of asymptotic expressions for general integrals has been considered by Luke (1969). Marichev (1983) states asymptotic results for integrals with complex parameters that apply specifically to Mellin-Barnes integrals with unity coefficients of the complex variable in the gamma functions. These results are generalized here to allow asymptotic series to be written for Mellin-Barnes integrals in which the coefficients of the complex variable are not unity. Also, the results are given in a form that is directly applicable to the evaluation of integrals encountered in turbulence problems.
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