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Chapter 5:
Integral Evaluation with Mellin Transforms
In this chapter I continue the development of techniques for evaluating the wavenumber integrals in turbulence propagation theory. With appropriate normalization, the wavenumber (κ) integration can be expressed in a standard form depending only on zero, one, or more parameters. If no parameters are present, the integration is performed simply by table lookup as was done in the last chapter. The one parameter case requires a transformation of the integral into the complex plane that is subsequently evaluated by pole-residue techniques. Details of this process are considered in this chapter. Integrals with more than one parameter are evaluated by extensions of this technique and are considered in Chap. 10. Tatarski considered the evaluation of one-parameter integrals in which inner scale was finite. He expanded the function that multiplies the decaying exponential, which contains the inner scale, into a Taylor series, and integrated term by term. Since the integral over each term of the power series converges absolutely, this method is valid. Tatarski expressed the resulting power series as a hypergeometric function. His approach does not always work with zero inner scale and restricts the range of problems that can be solved. This limitation does not apply to the technique discussed here. The integrals evaluated here are of the form I=∫ 0 ∞ dxx a f(bx c )g(dx e ), where f(x) and g(x) are Meijer's G-functions, a special case of which are generalized hypergeometric functions. The only restriction on a, b, c, d, and e is that the integral converges. Sometimes it will be required that d c /b e ≠1 .
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