Mellin-Transform Method for Integral Evaluation

George Fikioris was born in Boston, MA, on December 3, 1962. He received the Diploma of Electrical Engineering from the National Technical University of Athens, Greece (NTUA), in 1986, and the S.M. and Ph.D. degrees in Engineering Sciences from Harvard University in 1987 and 1993, respectively. From 1993 to 1998, he was an electronics engineer with the Air Force Research Laboratory, Hanscom AFB, MA. From 1999 to 2002, he was a researcher with the Institute of Communication and Computer Systems at the NTUA. From 2002 to February 2007, he was a lecturer at the school of Electrical and Computer Engineering, NTUA. In February 2007, he became an assistant professor at that school. He is the author or coauthor of over 25 papers in technical journals and numerous papers in conferences. Together with R. W. P. King and R. B. Mack, he has coauthored Cylindrical Antennas and Arrays, Cambridge University Press, 2002. His research interests include electromagnetics, antennas, and applied mathematics. Dr. Fikioris is a senior member of the IEEE (Antennas & Propagation, Microwave Theory & Techniques, and Education Societies), and a member of the American Mathematical Society and of the Technical Chamber of Greece.

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 methodis 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.

Introduction.- Mellin Transforms and theGamma Function.- Generalized Hypergeometric Functions, Meijer G-Functions, and Their Numerical Computation.- The Mellin-Transform Method of Evaluating Integrals.- Power Radiated by Certain Circular Antennas.- Aperture Admittance of a 2-D Slot Antenna.- An Integral Arising in the Theory of Biaxially Anisotropic Media.- On Closing the Contour.- Further Discussions.- Summary and Conclusions.