Mellin-transform Method for Integral Evaluation

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Morgan & Claypool Publishers, 2007 - Science - 67 pages
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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 In the third example, a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media is derived.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
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