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Tang K.-T. Mathematical Methods for Engineers and Scientists 1: Complex Analysis, Determinants and Matrices
Sрringеr, 2006. - 330 pages.
Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to make students comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.
Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student-oriented books. Topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations are presented in a discursive style that is readable and easy to follow.

Contents.
Part I Complex Analysis.
1 Complex Numbers.
Our Number System.
Addition and Multiplication of Integers.
Inverse Operations.
Negative Numbers.
Fractional Numbers.
Irrational Numbers.
Imaginary Numbers.
Logarithm.
Napier’s Idea of Logarithm.
Briggs’ Common Logarithm.
A Peculiar Number Called e.
The Unique Property of e.
The Natural Logarithm.
Approximate Value of e.
The Exponential Function as an Infinite Series.
Compound Interest.
The Limiting Process Representing e.
The Exponential Function ex.
Unification of Algebra and Geometry.
The Remarkable Euler Formula.
The Complex Plane.
Polar Form of Complex Numbers.
Powers and Roots of Complex Numbers.
Trigonometry and Complex Numbers.
Geometry and Complex Numbers.
Elementary Functions of Complex Variable.
Exponential and Trigonometric Functions of z.
Hyperbolic Functions of z.
Logarithm and General Power of z.
Inverse Trigonometric and Hyperbolic Functions.
Exercises.
2 Complex Functions.
Analytic Functions.
Complex Function as Mapping Operation.
Differentiation of a Complex Function.
Cauchy–Riemann Conditions.
Cauchy–Riemann Equations in Polar Coordinates.
Analytic Function as a Function of z Alone.
Analytic Function and Laplace’s Equation.
Complex Integration.
Line Integral of a Complex Function.
Parametric Form of Complex Line Integral.
Cauchy’s Integral Theorem.
Green’s Lemma.
Cauchy–Goursat Theorem.
Fundamental Theorem of Calculus.
4 Consequences of Cauchy’s Theorem.
Principle of Deformation of Contours.
The Cauchy Integral Formula.
Derivatives of Analytic Function.
Exercises.
3 Complex Series and Theory of Residues.
A Basic Geometric Series.
Taylor Series.
The Complex Taylor Series.
Convergence of Taylor Series.
Analytic Continuation.
Uniqueness of Taylor Series.
Laurent Series.
Uniqueness of Laurent Series.
Theory of Residues.
Zeros and Poles.
Definition of the Residue.
Methods of Finding Residues.
Cauchy’s Residue Theorem.
Second Residue Theorem.
Evaluation of Real Integrals with Residues.
Integrals of Trigonometric Functions.
Improper Integrals I: Closing the Contour with a Semicircle at Infinity.
Fourier Integral and Jordan’s Lemma.
Improper Integrals II: Closing the Contour with Rectangular and Pie-shaped Contour.
Integration Along a Branch Cut.
Principal Value and Indented Path Integrals.
Exercises.
Part II Determinants and Matrices.
4 Determinants.
Systems of Linear Equations.
Solution of Two Linear Equations.
Properties of Second-Order Determinants.
Solution of Three Linear Equations.
General Definition of Determinants.
Notations.
Definition of a nth Order Determinant.
Minors, Cofactors.
Laplacian Development of Determinants by a Row (or a Column).
Properties of Determinants.
Cramer’s Rule.
Nonhomogeneous Systems.
Homogeneous Systems.
Block Diagonal Determinants.
Laplacian Developments by Complementary Minors.
Multiplication of Determinants of the Same Order.
Differentiation of Determinants.
Determinants in Geometry.
Exercises.
5 Matrix Algebra.
Matrix Notation.
Definition.
Some Special Matrices.
Matrix Equation.
Transpose of a Matrix.
Matrix Multiplication.
Product of Two Matrices.
Motivation of Matrix Multiplication.
Properties of Product Matrices.
Determinant of Matrix Product.
The Commutator.
Systems of Linear Equations.
Gauss Elimination Method.
Existence and Uniqueness of Solutions of Linear Systems.
Inverse Matrix.
Nonsingular Matrix.
Inverse Matrix by Cramer’s Rule.
Inverse of Elementary Matrices.
Inverse Matrix by Gauss–Jordan Elimination.
Exercises.
6 Eigenvalue Problems of Matrices.
Eigenvalues and Eigenvectors.
Secular Equation.
Properties of Characteristic Polynomial.
Properties of Eigenvalues.
Some Terminology.
Hermitian Conjugation.
Orthogonality.
Gram–Schmidt Process.
Unitary Matrix and Orthogonal Matrix.
Unitary Matrix.
Properties of Unitary Matrix.
Orthogonal Matrix.
Independent Elements of an Orthogonal Matrix.
Orthogonal Transformation and Rotation Matrix.
Diagonalization.
Similarity Transformation.
Diagonalizing a Square Matrix.
Quadratic Forms.
Hermitian Matrix and Symmetric Matrix.
Definitions.
Eigenvalues of Hermitian Matrix.
Diagonalizing a Hermitian Matrix.
Simultaneous Diagonalization.
Normal Matrix.
Functions of a Matrix.
Polynomial Functions of a Matrix.
Evaluating Matrix Functions by Diagonalization.
The Cayley–Hamilton Theorem.
Exercises.
References.
Index.
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