# Mellin-transform Method for Integral Evaluation

Morgan & Claypool Publishers, 2007 - Science - 67 pages
This book introduces the Mellin-transform method for the exact calculation of one-dimensional definite integrals, and illustrates the application if this method to electromagnetics problems. Once the basics have been mastered, one quickly realizes that the method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the present method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. Two functions, the generalized hypergeometric function and the Meijer G-function, are very much related to the Mellin-transform method and arise frequently when the method is applied. Because these functions can be automatically handled by modern numerical routines, they are now much more useful than they were in the past. The Mellin-transform method and the two aforementioned functions are discussed first. Then the method is applied in three examples to obtain results, which, at least in the antenna/electromagnetics literature, are believed to be new. In the first example, a closed-form expression, as a generalized hypergeometric function, is obtained for the power radiated by a constant-current circular-loop antenna. The second example concerns the admittance of a 2-D slot antenna. In both these examples, the exact closed-form expressions are applied to improve upon existing formulas in standard antenna textbooks. In the third example, a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media is derived. Additional examples are also briefly discussed.

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### Contents

 Introduction 1 Mellin Transforms and the Gamma Function 5 Parseval Formula and Related Properties 7 24 Gamma Function 8 25 PsiFunction 9 26 Pochhammers Symbol 11 28 Table Lookup of Mellin Transforms MellinBarnes Integrals 13 Generalized Hypergeometric Functions Meijer GFunctions and Their Numerical Computation 17
 On Closing the Contour 39 Further Discussions 41 94 Significance of the Poles to the Right Asymptotic Expansions 42 96 Numerical Evaluation of Integrals by Modern Routines 43 97 Additional Reading 44 Summary and Conclusions 45 On the ConvergenceDivergence of Deﬁnite Integrals 47 A2 Rules for Determining ConvergenceDivergence 48

 32 Remarks 20 The MellinTransform Method of Evaluating Integrals 21 42 A First Example 22 Power Radiated by Certain Circular Antennas 25 52 CircularPatch Microstrip Antennas Cavity Model 26 53 Integral Evaluation 27 54 Application to Electrically Large Loop Antennas 28 Aperture Admittance of a 2D Slot Antenna 31 An Integral Arising in the Theory of Biaxially Anisotropic Media 35
 A3 Examples 51 The Lemma of Section 27 53 B2 Derivation of 238 54 Alternative Derivations or Verifications for the Integrals of Section 42 and Chapters 5 and 6 55 Additional Examples from the Electromagnetics Area 57 D2 An Integral Relevant to the ThinWire Loop Antenna 58 References 61 Author Biography 67 Copyright