Advanced Engineering Mathematics with MATLAB, Second Edition

This text/reference covers essential areas of engineering mathematics involving single, multiple, and complex variations. Taken as a whole, this book provides a succinct, carefully organized guide for mastering engineering mathematics. Unlike typical textbooks, Advanced Engineering Mathematics begins with a thorough exploration of complex variables because they provide powerful techniques for understanding topics, such as Fourier, Laplace and z-transforms, introduced later in the text. The book contains a wealth of examples, both classic problems used to illustrate concepts, and interesting real-life examples from scientific literature. Ideal for a two-semester course on advanced engineering mathematics, Advanced Engineering Mathematics is concise and well-organized, unlike the long, detailed texts used to teach this subject. Since almost every engineer and many scientists need the skills covered in this book for their daily work, Advanced Engineering Mathematics also makes an excellent reference for practicing engineers and scientists.

Table of Contents:
Introduction Complex Variables Complex Numbers Finding Roots The Derivative in the Complex Plane: The Cauchy-Riemann Equations Line Integrals Cauchy-Goursat Theorem Cauchy's Integral Formula Taylor and Laurent Expansions and Singularities Theory of Residues Evaluation of Real Definite Integrals Fourier Series Fourier Series Properties of Fourier Series Half-Range Expansions Fourier Series with Phase Angles Complex Fourier Series The Use of Fourier Series in the Solution of Ordinary Differential Equations Finite Fourier Series The Fourier Transform Fourier Transform Fourier Transforms Containing the Delta Function Properties of Fourier Transforms Inversion of Fourier Transforms Convolution Solution of Ordinary Differential Equations by Fourier Transforms The Laplace Transform Definition and Elementary Properties Heaviside Step and Dirac Delta Functions Some Useful Theorems The Laplace Transform of a Periodic Function Inversion by Partial Fractions: Heaviside's Expansion Theorem Convolution Integral Equations Solution of Linear Differential Equations with Constant Coefficients Transfer Functions, Green's Function, and Indicial Admittance Inversion by Contour Integration The Z-Transform The Relationship of the Z-Transform to the Laplace Transform Some Useful Properties Inverse Z-Transforms Solution of Difference Equations Stability of Discrete-Time Systems The Sturm-Liouville Problem Eigenvalues and Eigenfunctions Orthogonality of Eigenfunctions Expansion in Series of Eigenfunction A Singular Sturm-Liouville Problem: Legendre's Equation Another Singular Sturm-Liouville Problem: Bessel's Equation The Wave Equation The Vibrating String Initial Conditions: Cauchy Problem Separation of Variables D'Alembert's Formula The Laplace Transform Method Numerical Solution of the Wave Equation The Heat Equation Derivation of the Heat Equation Initial and Boundary Conditions Separation of Variables The Laplace Transform Method The Fourier Transform Method The Superposition Integral Numerical Solution of the Heat Equation Laplace's Equation Derivation of Laplace's Equation Boundary Conditions Separation of Variables The Solution of Laplace's Equation on the Upper Half-Plane Poisson's Equation in a Rectangle The Laplace Transform Method Numerical Solution of Laplace's Equation Vector Analysis Review Divergence and Curl Line Integrals The Potential Function Surface Integrals Green's Lemma Stokes' Theorem Divergence Theorem Linear Algebra Fundamentals of Linear Equations Determinants Cramer's Rule Row Echelon Form and Gaussian Elimination Eigenvalues and Eigenvectors Systems of Linear Differential Equations Answers to the Odd-Numbered Problems Index