Applied Complex Analysis with Partial Differential Equations

Applied Complex Analysis with Partial Differential Equations

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For Introductory courses on Complex Analysis or Complex Analysis and Partial Differential Equations for engineering, physics, and mathematics students with a calculus background.This student-friendly text presents traditional material using a modern approach that invites the use of technology. Abundant exercises, examples and graphics make this text a comprehensive and visually appealing resource for students, as well as a quick reference tool for engineers and physicists.show more

Product details

  • Hardback | 883 pages
  • 212.1 x 244.1 x 40.6mm | 1,632.95g
  • Pearson Education (US)
  • Pearson
  • United States
  • English
  • references, index
  • 0130892394
  • 9780130892393

About Nakhle H. Asmar

Nakhle H. Asmar received his Ph.D in mathematics from the University of Washington in 1986. After spending two years on the faculty of California State University, Long Beach, he joined the University of Missouri, Columbia in 1988, where he is currently Professor of Mathematics. He is the author of the book "Partial Differential Equations and Boundary Value Problems," published by Prentice Hall in 1999. He is also the author or co-author of over forty research articles in the areas of harmonic analysis, Fourier series, and functional analysis. His research received support from the National Science Foundation (U.S.A.). He has received several teaching awards from the University of Missouri, including the William T. Kemper Fellowship Award, the Arts and Science Student Government Purple Chalk Award, and the Provost's Outstanding Junior Faculty Teaching Award. He is a member of the Research Board of the University of Missouri and a member of the College of Reviewers for the Canada Research Chairs program. The author can be contacted by e-mail at the following address:nakhle@math.missouri.edushow more

Table of contents

1. Complex Numbers and Functions. Complex Numbers. The Complex Plane. Polar Form. Complex Functions. The Complex Exponential. Trigonometric and Hyperbolic Functions. Logarithms and Powers.2. Analytic Functions. Regions of the Complex Plane. Limits and Continuity. Analytic Functions. The Cauchy-Riemann Equations. Harmonic Functions and Laplace's Equation. Supplement on Calculus of Functions of Several Variables. Differentiation of Functions of Several Variables.3. Complex Integration. Contours and Paths in the Complex Plane. Complex Integration. Independence of Path. Cauchy's Integral Theorem. Proof of Cauchy's Integral Theorem. Cauchy's Integral Formula. Bounds for Moduli of Analytic Functions. Applications to Harmonic Functions. Goursat's Theorem.4. Complex Series. Sequences and Series of Complex Numbers. Sequences and Series of Functions. Power Series. Taylor Series. Laurent Series. Zeros and Poles. Harmonic Functions and Fourier Series.5. Residue Theory. Cauchy's Residue Theorem. Definite Integrals of Trigonometric Functions. Improper Integrals Involving Rational and Exponential Functions. Improper Integrals of Products of Rational and Trigonometric Functions. Advanced Integrals by Residues. Summing Series by Residues. The Counting Theorem and Rouche's Theorem.6. Conformal Mappings. Basic Properties. Linear Fractional Transformations. Solving Dirichlet Problems with Conformal Mappings. The Schwarz-Christoffel Transformation. Green's Functions. Poisson's Equation and Neumann Problems.7. Fourier Series. Periodic Functions. Fourier Series. Fourier Series With Arbitrary Period. Half-Range Expansions. Complex Form of Fourier Series. Proof of the Fourier Representation Theorem.8. Partial Differential Equations in Rectangular Coordinates. Partial Differential Equations in Physics and Engineering. Solution of the One Dimensional Wave Equation: The Method of Separation of Variables. The One Dimensional Heat Equation. Heat Conduction in Bars: Varying the Boundary Conditions. Two Dimensional Wave and Heat Equations. Laplace's Equation in Rectangular Coordinates. The Method of Eigenfunction Expansions. Neumann Problems and Robin Conditions.9. Partial Differential Equations in Polar and Cylindrical Coordinates. Laplace's Equation in Polar Coordinates. Vibrations of a Circular Membrane: Symmetric Case. Vibrations of a Circular Membrane: General Case. Steady-State Temperature in a Cylinder. The Helmholtz and Poisson Equations. Supplement on Bessel Functions and Series Expansions. Bessel's Equation and Bessel Functions. Bessel Series Expansions.10. Partial Differential Equations in Spherical Coordinates. Preview of Problems and Methods. Dirichlet Problems with Symmetry. Spherical Harmonics and the General Dirichlet Problem. The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations. Supplement on Legendre Functions and Series Expansions. Legendre's Differential Equation and Legendre Polynomials. Legendre Series. Associated Legendre Functions and Series Expansions.11. The Fourier Transform and its Applications. The Fourier Transform. Operational Properties. The Fourier Transform Method. The Heat Equation and Gauss's Kernel. The Poisson Integral and the Hilbert Transform. The Fourier Cosine and Sine Transforms. Problems Involving Semi-Infinite Intervals.12. The Laplace and Hankel Transforms with Applications. The Laplace Transform. Further Properties of the Laplace Transform. The Laplace Transform Method. The Hankel Transform with Applications.Appendix A. Ordinary Differential Equations: Review of Concepts and Methods. Linear Ordinary Differential Equations. Linear Ordinary Differential Equations with Constant Coefficients. Methods for Solving Ordinary Differential Equations. The Method of Power Series. The Method of Frobenius.Appendix B. Tables of Transforms. Fourier Transforms. Fourier Cosine Transforms. Fourier Sine Transforms. Laplace Transforms.Bibliography. Answers to Selected Exercises. Index.show more

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