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E-grāmata: Mathematical Methods for Engineers and Scientists 1: Complex Analysis and Linear Algebra

  • Formāts: PDF+DRM
  • Izdošanas datums: 25-Oct-2022
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783031056789
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Mathematical Methods for Engineers and Scientists 1: Complex Analysis and Linear AlgebraPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 25-Oct-2022
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783031056789
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Part 1 of this popular graduate-level textbook focuses on mathematical methods involving complex analysis, determinants, and matrices, including updated and additional material covering conformal mapping. The second edition comes with extensive updates and additions, making them a more complete reference for graduate science and engineering students while imparting comfort and confidence in using advanced mathematical tools in both upper-level undergraduate and beginning graduate courses. This set of student-centered textbooks presents topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transformations, and ordinary and partial differential equations in a discursive style that is clear, engaging, and easy to follow. Replete with pedagogical insights from an author with more than 30 years of experience in teaching applied mathematics, this indispensable set of books features numerous clearly stated and completely worked out examples together with carefully selected problems and answers that enhance students' understanding and analytical skills.
Part I Complex Analysis
1 Complex Numbers
3(62)
1.1 Our Number System
3(10)
1.1.1 Addition and Multiplication of Integers
4(1)
1.1.2 Inverse Operations
5(1)
1.1.3 Negative Numbers
6(1)
1.1.4 Fractional Numbers
7(1)
1.1.5 Irrational Numbers
8(1)
1.1.6 Imaginary Numbers
9(4)
1.2 Logarithm
13(5)
1.2.1 Napier's Idea of Logarithm
13(2)
1.2.2 Briggs' Common Logarithm
15(3)
1.3 A Peculiar Number Called e
18(4)
1.3.1 The Unique Property of e
18(1)
1.3.2 The Natural Logarithm
19(2)
1.3.3 Approximate Value of e
21(1)
1.4 The Exponential Function as an Infinite Series
22(3)
1.4.1 Compound Interest
22(2)
1.4.2 The Limiting Process Representing e
24(1)
1.4.3 The Exponential Function e*
24(1)
1.5 Unification of Algebra and Geometry
25(4)
1.5.1 The Remarkable Euler Formula
25(1)
1.5.2 The Complex Plane
26(3)
1.6 Polar Form of Complex Numbers
29(19)
1.6.1 Powers and Roots of Complex Numbers
31(3)
1.6.2 Trigonometry and Complex Numbers
34(7)
1.6.3 Geometry and Complex Numbers
41(7)
1.7 Elementary Functions of Complex Variable
48(17)
1.7.1 Exponential and Trigonometric Functions of z
48(2)
1.7.2 Hyperbolic Functions of z
50(2)
1.7.3 Logarithm and General Power of z
52(6)
1.7.4 Inverse Trigonomeric and Hyperbolic Functions
58(3)
Exercises
61(4)
2 Complex Functions
65(50)
2.1 Analytic Functions
65(22)
2.1.1 Complex Function as Mapping Operation
66(1)
2.1.2 Differentiation of a Complex Function
67(2)
2.1.3 Cauchy-Riemann Conditions
69(3)
2.1.4 Cauchy-Riemann Equations in Polar Coordinates
72(2)
2.1.5 Analytic Function as a Function of z Alone
74(5)
2.1.6 Analytic Function and Laplace's Equation
79(8)
2.2 Complex Integration
87(5)
2.2.1 Line Integral of a Complex Function
87(2)
2.2.2 Parametric Form of Complex Line Integral
89(3)
2.3 Cauchy's Integral Theorem
92(7)
2.3.1 Green's Lemma
92(2)
2.3.2 Cauchy-Goursat Theorem
94(2)
2.3.3 Fundamental Theorem of Calculus
96(3)
2.4 Consequences of Cauchy's Theorem
99(16)
2.4.1 Principle of Deformation of Contours
99(1)
2.4.2 The Cauchy Integral Formula
100(2)
2.4.3 Derivatives of Analytic Function
102(7)
Exercises
109(6)
3 Complex Series and Theory of Residues
115(66)
3.1 A Basic Geometric Series
115(1)
3.2 Taylor Series
116(9)
3.2.1 The Complex Taylor Series
116(1)
3.2.2 Convergence of Taylor Series
117(2)
3.2.3 Analytic Continuation
119(2)
3.2.4 Uniqueness of Taylor Series
121(4)
3.3 Laurent Series
125(11)
3.3.1 Uniqueness of Laurent Series
129(7)
3.4 Theory of Residues
136(14)
3.4.1 Zeros and Poles
136(1)
3.4.2 Definition of the Residue
137(1)
3.4.3 Methods of Finding Residues
138(4)
3.4.4 Cauchy's Residue Theorem
142(1)
3.4.5 Second Residue Theorem
143(7)
3.5 Evaluation of Real Integrals with Residues
150(31)
3.5.1 Integrals of Trigonometric Functions
150(4)
3.5.2 Improper Integrals I: Closing the Contour with a Semicircle at Infinity
154(3)
3.5.3 Fourier Integral and Jordan's Lemma
157(6)
3.5.4 Improper Integrals II: Closing the Contour with Rectangular and Pie-Shaped Contour
163(5)
3.5.5 Integration Along a Branch Cut
168(3)
3.5.6 Principal Value and Indented Path Integrals
171(5)
Exercises
176(5)
4 Conformal Mapping
181(112)
4.1 Examples of Problems Solved by Conformal Mappings
182(23)
4.2 Invariance of the Laplace Equation
205(3)
4.3 Conformal Mapping
208(4)
4.4 Complex Potential
212(6)
4.5 Flow of Fluids
218(14)
4.5.1 Irrotation Flow and Velocity Potential
219(3)
4.5.2 Incompressibility of the Fluid
222(2)
4.5.3 Stream Function and Stream Lines
224(5)
4.5.4 Complex Velocity
229(3)
4.6 The Joukowski Transformation
232(7)
4.6.1 Properties of the Joukowski Transformation
233(2)
4.6.2 Joukowski Profiles
235(4)
4.7 The Mcibius Transformation
239(35)
4.7.1 Linear Transformation
240(3)
4.7.2 Inversion
243(12)
4.7.3 General Bilinear Transformation
255(6)
4.7.4 Some Properties of Bilinear Transformation
261(3)
4.7.5 Mapping Two Distinct Circles into Concentric Circles
264(10)
4.8 The Schwarz-Christoffel Transformation
274(19)
4.8.1 Formulation of the Schwarz-Christoffel Transformation
275(4)
4.8.2 Convention Regarding the Polygon
279(1)
4.8.3 Examples
280(5)
Exercises
285(8)
Part II Linear Algebra
5 Linear Algebra and Vector Space
293(44)
5.1 Linear System of Equations
294(13)
5.1.1 Cramer's Rule
294(3)
5.1.2 Gaussian Elimination
297(5)
5.1.3 LU Decomposition
302(3)
5.1.4 Geometry and Linear Equations
305(2)
5.2 Vector Space
307(30)
5.2.1 Definition of Vector Space
307(2)
5.2.2 Dot Product and Length of a Vector
309(2)
5.2.3 Essence of Finite Dimensional Vector Space
311(17)
5.2.4 Infinite Dimensional Function Space
328(2)
Exercises
330(7)
6 Determinants
337(42)
6.1 Systems of Linear Equations
337(6)
6.1.1 Solution of Two Linear Equations
337(2)
6.1.2 Properties of Second-Order Determinants
339(1)
6.1.3 Solution of Three Linear Equations
340(3)
6.2 General Definition of Determinants
343(9)
6.2.1 Notations
343(3)
6.2.2 Definition of a nth-Order Determinant
346(2)
6.2.3 Minors, Cofactors
348(1)
6.2.4 Laplacian Development of Determinants by a Row (or a Column)
349(3)
6.3 Properties of Determinants
352(6)
6.4 Cramer's Rule
358(3)
6.4.1 Nonhomogeneous Systems
358(2)
6.4.2 Homogeneous Systems
360(1)
6.5 Block Diagonal Determinants
361(3)
6.6 Laplacian Developments by Complementary Minors
364(3)
6.7 Multiplication of Determinants of the Same Order
367(2)
6.8 Differentiation of Determinants
369(1)
6.9 Determinants in Geometry
370(9)
Exercises
374(5)
7 Matrix Algebra
379(44)
7.1 Matrix Notation
379(7)
7.1.1 Definition
379(1)
7.1.2 Some Special Matrices
380(2)
7.1.3 Matrix Equation
382(2)
7.1.4 Transpose of a Matrix
384(2)
7.2 Matrix Multiplication
386(14)
7.2.1 Product of Two Matrices
386(4)
7.2.2 Motivation of Matrix Multiplication
390(1)
7.2.3 Properties of Product Matrices
391(5)
7.2.4 Determinant of Matrix Product
396(2)
7.2.5 The Commutator
398(2)
7.3 Systems of Linear Equations
400(8)
7.3.1 Gauss Elimination Method
401(3)
7.3.2 Existence and Uniqueness of Solutions of Linear Systems
404(4)
7.4 Inverse Matrix
408(15)
7.4.1 Non-singular Matrix
408(1)
7.4.2 Inverse Matrix by Cramer's Rule
409(4)
7.4.3 Inverse of Elementary Matrices
413(2)
7.4.4 Inverse Matrix by Gauss-Jordan Elimination
415(2)
Exercises
417(6)
8 Eigenvalue Problems of Matrices
423(62)
8.1 Eigenvalues and Eigenvectors
423(12)
8.1.1 Secular Equation
423(8)
8.1.2 Properties of Characteristic Polynomial
431(3)
8.1.3 Properties of Eigenvalues
434(1)
8.2 Some Terminology
435(4)
8.2.1 Hermitian Conjugation
435(1)
8.2.2 Orthogonality
436(2)
8.2.3 Gram-Schmidt Process
438(1)
8.3 Unitary Matrix and Orthogonal Matrix
439(8)
8.3.1 Unitary Matrix
439(1)
8.3.2 Properties of Unitary Matrix
440(1)
8.3.3 Orthogonal Matrix
441(2)
8.3.4 Independent Elements of an Orthogonal Matrix
443(1)
8.3.5 Orthogonal Transformation and Rotation Matrix
444(3)
8.4 Diagonalization
447(9)
8.4.1 Similarity Transformation
447(3)
8.4.2 Diagonalizing a Square Matrix
450(3)
8.4.3 Quadratic Forms
453(3)
8.5 Hermitian Matrix and Symmetric Matrix
456(12)
8.5.1 Definitions
456(1)
8.5.2 Eigenvalues of Hermitian Matrix
456(2)
8.5.3 Diagonalizing a Hermitian Matrix
458(8)
8.5.4 Simultaneous Diagonalization
466(2)
8.6 Normal Matrix
468(2)
8.7 Functions of a Matrix
470(15)
8.7.1 Polynomial Functions of a Matrix
470(2)
8.7.2 Evaluating Matrix Functions by Diagonalization
472(4)
8.7.3 The Cayley-Hamilton Theorem
476(4)
Exercises
480(5)
References 485(2)
Index 487
K.T. Tang received his B.S. in Engineering Physics and M.A. in Mathematics from the University of Washington and his Ph.D. in Physics from Columbia University. He did postdoctoral studies in Chemistry at Berkeley and Harvard. He worked as an engineer at Collins Radio Company and the Boeing Company. Dr. Tang regards teaching as his calling, although his research accomplishments are also considerable. He authored/co-authored over 150 research papers in professional journals. He lectured widely in Asia, Europe, and North America. He is a recipient of a Distinguished U.S. Senior Scientist Award from the Alexander von Humboldt Foundation. He also received a Faculty Excellence Award and a Presidential Medal from Pacific Lutheran University, where he is a professor emeritus of Physics. He is a fellow of the American Physical Society.
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