The Cauchy Method of Residues, Softcover reprint of the original 1st ed. 1993
Volume 2: Theory and Applications
Mathematics and Its Applications Series, Vol. 259

Volume 1, i. e. the monograph The Cauchy Method of Residues - Theory and Applications published by D. Reidel Publishing Company in 1984 is the only book that covers all known applications of the calculus of residues. They range from the theory of equations, theory of numbers, matrix analysis, evaluation of real definite integrals, summation of finite and infinite series, expansions of functions into infinite series and products, ordinary and partial differential equations, mathematical and theoretical physics, to the calculus of finite differences and difference equations. The appearance of Volume 1 was acknowledged by the mathematical community. Favourable reviews and many private communications encouraged the authors to continue their work, the result being the present book, Volume 2, a sequel to Volume 1. We mention that Volume 1 is a revised, extended and updated translation of the book Cauchyjev raeun ostataka sa primenama published in Serbian by Nau~na knjiga, Belgrade in 1978, whereas the greater part of Volume 2 is based upon the second Serbian edition of the mentioned book from 1991. Chapter 1 is introductory while Chapters 2 - 6 are supplements to the corresponding chapters of Volume 1. They mainly contain results missed during the preparation of Volume 1 and also some new results published after 1982. Besides, certain topics which were only briefly mentioned in Volume 1 are treated here in more detail.

1. Introduction.- 1.1. Organization and References.- 1.2. Errata for Volume 1.- 1.3. Notations, Definitions and Theorems.- 2. Evaluation of Residues.- 3. Applications of Calculus of Residues in the Theory of Functions.- 3.1. A Generalization of the Principle of the Argument.- 3.2. Runge’s Phenomenon.- 3.3. Expansion into Bürmann’s Series.- 3.4. Carleman’s Theorem.- 3.5. Analytic Continuation of Cauchy Type Integrals.- 3.6. An Asymptotic Formula.- 3.7. Miscellaneous Applications.- 4. Evaluation of Real Definite Integrals by Means of Residues.- 4.1. Integrals with Infinite Limits.- 4.2. Integrals with Finite Limits.- 4.3. ?ebyšev’s Approximation of the Integral of a Positive Function.- 4.4. A Note on some Papers of Ostrogradski and Bouniakowski.- 5. Evaluation of Finite and Infinite Sums by Residues.- 5.1. Gauss’ Sums.- 5.2. The Riemann Zeta Function.- 5.3. Miscellaneous Summations.- 6. Applications of Calculus of Residues to Special Functions.- 6.1. Polygamma Functions of Arbitrary Order.- 6.2. A Connection Between the Exponential and the Gamma Function.- 6.3. Residues of Some Functions Related to the Gamma Function.- 6.4. Some Integrals Involving the Gamma Function.- 7. Master’s dissertation of J. V. Sohocki.- 7.1. Introduction.- 7.2. Properties of Residues.- 7.3. Two Formulas of Lagrange.- 7.4. Continued Fractions.- 7.5. Legendre’s Polynomials.- 7.6. Expansion of a Function by Means of Continued Fractions.- 8. On the Principal and the Generalized Value of Improper Integrals.- 8.1. Substitution in Complex Integrals.- 8.2. The Principal Value for Higher Order Poles.- 8.3. The Principal Value in the Case when the Limits of Integration are Singular Points.- 8.4. Generalized Value of an Improper Integral with Infinite Limits.- 8.5. Generalized Value of anImproper Integral Between Finite Limits.- 9. Applications of the Calculus of Residues to Numerical Evaluation of Integrals.- 10. Inclusive Calculus of Residues.- 11. Complex Polynomials Orthogonal on the Semicircle.- 11.1. Introduction.- 11.2. Orthogonality on the Semicircle.- 11.3. Existence and Representation of ?n.- 11.4. Recurrence Relation.- 11.5. Jacobi Weight.- 11.6. Symmetric Weights and Gegenbauer Weights.- 11.7. The zeros of ?n(Z).- 12. A Representation of Half Plane Meromorphic Functions.- 13. Calculus of Residues and Distributions.- 13.1. Test Functions and Distributions.- 13.2. The Spaces D and D’.- 13.3. The Spaces E and E’.- 13.4. The Spaces $$ {\mathcal{O}_a} $$ And $$ {\mathcal{O}_a}' $$.- 13.5. A Distributional Representation of Half Plane Meromorphic Functions.- 13.6. A Generalization of the Residue Theorem.- 13.7. A Generalization of the Cauchy Integral Theorem for an Infinite Strip.- Name Index.