The Cauchy Method of Residues

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. ?ebysev'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.