Functions of a complex variable. (2015)

• Record Type: Book
• Further Information: Note: Hemant Kumar Pathak, Ravi Agarwal, Yeol Je Cho.
• Authors: Pathak, Hemant Kumar; Agarwal, Ravi P; Cho, Yeol Je
• Contents: Complex Numbers and Their Geometrical Representation; Introduction; Complex Numbers; Modulus and Argument of Complex Numbers; Geometrical Representations of Complex Numbers; Modulus and Argument of Complex Numbers; Properties of Moduli; Properties of Arguments; Equations of Straight Lines; Equations of Circles; Inverse Points; Relations between Inverse Points with Respect to Circles; Riemann Spheres and Point at Infinity; Cauchy–Schwarz’s Inequality and Lagrange’s Identity; Historical Remarks Analytic Functions; Metric Spaces and Topology of C; Functions of Complex Variables; Uniform Continuity; Differentiability; Analytic and Regular Functions; Cauchy–Riemann Equations; Methods of Constructing Analytic Functions; Historical Remarks Power Series and Elementary Functions ; Power Series; Certain Theorems on Power Series; Elementary Functions of a Complex Variable; Many-Valued Functions: Branches; Logarithms and Power Functions; The Riemann Surfaces for Log z; Historical Remarks Conformal Representation; Mappings or Transformations; Jacobian of Transformations; Conformal Mappings; Sufficient Condition for w = f(z) to Represent Conformal Mappings; Necessary Conditions for w = f(z) to Represent Conformal Mappings; Superficial Magnification; Some Elementary Transformations; Linear Transformations; Bilinear or Möbius Transformations; Product or Resultant of Two Bilinear Transformations; Every Bilinear Transformation Is the Resultant of Elementary Transformations; Bilinear Transformation as the Resultant of an Even Number of Inversions; The Linear Groups; Cross-Ratios; Preservation of Cross-Ratio under Bilinear Transformations; Preservation of the Family of Circles and Straight Lines under Bilinear Transformations; Two Important Families of Circles; Fixed Point of Bilinear Transformations; Normal Form of a Bilinear Transformation; Elliptic, Hyperbolic, and Parabolic Transformations; Special Bilinear Transformations; Historical Remarks Special Transformations; Introduction; The Transformation w = za Where a Is a Complex Number; The Inverse Transformation z = √w; The Exponential Transformation w = ez ; The Logarithmic Transformation w = log z ; The Trigonometrical Transformation z = c sin w ; The Transformation w = tan z ; The Transformation w = tan2 (π/4a√z ); The Transformation w = 1/2 (z + 1/z ); The Transformation z = 1/2 (w + 1/w ); Historical Remarks Complex Integrations; Complex Integrations; Complex Integrals; Cauchy's Theorem; Indefinite Integrals of Primitives; Cauchy's Integral Formula; Derivatives of Analytic Functions; Higher-Order Derivatives; Morera’s Theorem; Poisson's Integral Formula for Circles; Cauchy's Inequality; Liouville’s Theorem; Cauchy's Theorem and Integral Formulas; The Homotopic Version of Cauchy's Theorem and Simple Connectivity; Expansion of Analytic Functions as Power Series; Historical Remarks Singularities; Zeros of Analytic Functions; Singular Points; The "Point at Infinity"; Characterization of Polynomials; Characterization of Rational Functions Residue Theory and Principle of Argument; Introduction; The Residues at Singularities; Calculation of Residues in Some Special Cases; Residues at Infinity; Some Residue Theorems; Argument Principle and Rouché’s Theorem; Schwarz’s Lemma; The Inverse Functions; Formulas of Poisson, Hilbert, and Bromwich Calculus of Residues; Evaluations of Definite Integrals by Contour Integrations; Integrations around the Unit Circle; Evaluations of Integrals of Type ∫∞ -∞ f(x) dx ; Jordan's Inequality; Jordan's Lemma; Evaluations of Integrals of Forms; Cases of Poles on the Real Axis; Cases of Poles on the Real Axis (Indenting Method); Integrals of Many-Valued Functions; Quadrants or Sectors of Circles as Contours; Rectangular Contours Uniform Convergence; Uniform Convergence of Sequences; Uniform Convergence of Series; Hardy's Tests for Uniform Convergence; Continuity of Sum Functions of Series; Term-by-Term Integrations; Analyticity of Sum Functions of Series (Term-by-Term Differentiations); Uniform Convergence of Power Series; Formulas of Parseval, Schwarz, and Poisson; Functions Defined by Integrals Harmonic Functions ; Harmonic Functions; Inverse Mappings and Univalent Functions; Global Mapping Theorem; Riemann's Mapping Theorem; Historical Remarks Analytic Continuation; Introduction; Analytic Continuation; Power Series Methods of Analytic Continuation Answers to Selected Questions Bibliography Index
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2015
• Extent: 1 online resource, illustrations (black and white)
• Subjects: 515.9; Functions of complex variables
• Languages: English
• ISBNs: 9781498720182; 9781498720168; 9781498720175
• Related ISBNs: 9781498720151
• Notes: Note: Description based on CIP data; item not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.