## Mellin-transform Method for Integral EvaluationThis 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 In the third example, a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media is derived.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 |

67 | |

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### Common terms and phrases

A. P. Prudnikov aforementioned analytic continuation aperture Appendix Application applying the Mellin-transform approximation ascending series asymptotic expansion Bessel function Brychkov calculations Chapt Chapters 2–4 circular loop close the contour closed-form expression coefﬁcients contour at left convergence/divergence converges if Re{e deﬁned deﬁnite integrals derivation double poles electromagnetics Entry exact results ﬁeld Fikioris ﬁnal result ﬁnd ﬁnite ﬁrst example Fourier gamma function hypergeometric function inﬁnite integral arising integral converges Integral Evaluation integrand integration interval Laplace transform lattice left poles lemma Math Mathematica Meijer G-function Mellin convolution Mellin transform Mellin-transform method Mellin–Barnes integral representation Microstrip Antennas numerical evaluation O. I. Marichev obtain packaged routines parameters pFq and G r(z+ Re{z recurrence formula 2.16 reduction table residue right poles right-hand side Section 2.7 signiﬁcance simple poles special functions Symbolic routines table of integrals Taylor series transform method usual zero