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Chapter 11:
Integral Evaluation with N Parameters
The mathematical technique developed in the last chapter is used to find analytical solutions to seven problems. The first three are purely mathematical problems, and the last four are problems related to turbulence. The first three examples have solutions that are in integral tables. The same solution and range of applicability is obtained with the Mellin transform method. The second problem, which contains a product of three Bessel functions, has applicability in the calculation of the correlation coefficients of various modes on a circular aperture. The third example has an integrand that contains a sinusoid and the product of Bessel functions. Two cases are considered; in the first there are three Bessel functions and in the second there are N. This example is the only one given in this book that illustrates how to use the technique in more than two complex planes. For the first four problems asymptotic solutions are not required. In the last three problems they are required. It may not be necessary to use asymptotic solutions for certain applications since if enough Taylor series terms are used they can provide a sufficiently accurate answer. In the calculation of the power spectral density in Sec. 6.7 a 40-term Taylor series was sufficient to calculate the spectrum to large frequencies. The asymptotic solution does have one advantage over the Taylor series in that the dependence on parameters is clearly evident. Four problems in wave propagation in turbulence are solved. These problems were chosen to illustrate the various techniques one must use to obtain asymptotic solutions under different conditions. The first three are generalizations of problems solved earlier in the book. It is shown that these solutions approach the former solutions at the proper limits. These three solutions determine the effect of outer scale on tilt anisoplanatism, the effect on tilt variance with both inner-scale and outer-scale effects present, and the tilt power spectral density with outer-scale effects. The fourth solution gives phase structure and correlation functions with inner and outer scale effects. These solutions demonstrate one of the most valuable assets of this technique. From the first term that couples the two effects, one can determine when the coupling starts to affect the result. From this term, one can determine when coupling can be neglected. The specific form of the coupling is difficult to ascertain when numerical methods are used to evaluate the integral.
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Bessel functions


Correlation function

Numerical analysis

Wave propagation

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