Riassunto:
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.
Contenuti:
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
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