Download Mellin - Transform Method for Integral Evaluation: by George Fikioris PDF
By George Fikioris
This ebook introduces the Mellin-transform technique for the precise calculation of one-dimensional yes integrals, and illustrates the applying if this system to electromagnetics difficulties. as soon as the fundamentals were mastered, one fast realizes that the tactic is very strong, usually yielding closed-form expressions very tricky to return up with different equipment or to infer from the standard tables of integrals. but, instead of different equipment, the current approach is particularly common to use; it always calls for onerous calculations, yet little ingenuity. features, the generalized hypergeometric functionality and the Meijer G-function, are greatly with regards to the Mellin-transform approach and come up often whilst the tactic is utilized. simply because those features should be instantly dealt with by way of glossy numerical workouts, they're now even more precious than they have been some time past. The Mellin-transform strategy and the 2 aforementioned capabilities are mentioned first. Then the strategy is utilized in 3 examples to acquire effects, which, at the least within the antenna/electromagnetics literature, are believed to be new. within the first instance, a closed-form expression, as a generalized hypergeometric functionality, is received for the ability radiated via a constant-current circular-loop antenna. the second one instance issues the admittance of a 2-D slot antenna. In either those examples, the precise closed-form expressions are utilized to enhance upon present formulation in normal antenna textbooks. within the 3rd instance, an easy expression for an fundamental coming up in contemporary, unpublished stories of unbounded, biaxially anisotropic media is derived. extra examples also are in brief mentioned"
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Extra info for Mellin - Transform Method for Integral Evaluation: Introduction and Applications to Electromagnetics
Sample text
When 0 < kb < 3, the error is seen in Fig. 1 to decrease monotonically and rapidly as kb decreases. book Mobk068 March 28, 2007 7:42 book Mobk068 March 28, 2007 7:42 35 CHAPTER 7 An Integral Arising in the Theory of Biaxially Anisotropic Media The unpublished studies [44], [45] (which are somewhat similar to the recent papers [46], [47]) deal with the Green’s function in unbounded, biaxially anisotropic media, with the aim of understanding the behavior of the two types of waves possible in such media.
For an infinite integration interval, it is not sufficient for the integrand to vanish at infinity for the integral to converge∗ ; the rate of this vanishing is also important. In this appendix, we provide simple, systematic rules of thumb that can help us determine when a given integral converges and apply these rules to a large number of examples. Our integration path is always on the real axis. 1 SOME REMARKS ON OUR RULES Before giving our rules, some clarifications are necessary. (i) In this appendix, and throughout this book, our integrals are “classical” ones.
It applies to integrals f (x) that are Mellin convolutions. 1) 0 where x is a positive parameter. Many integrals have this form (all Laplace transforms and all Fourier cosine transforms, for example), or can easily be written in terms of integrals having this form (all Fourier transforms, for example). We first give a general (but sketchy) description of the Mellin-transform method. 2) which will, hopefully, be an Mellin–Barnes integral. 13) ˜ − z). 5), in which f˜(z) = g˜ (z)h(1 ˜ − z), which must overlap.