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E-grāmata: Mathematical Methods for Engineers and Scientists 1: Complex Analysis, Determinants and Matrices

4.33/5 (4 ratings by Goodreads)
  • Formāts: PDF+DRM
  • Izdošanas datums: 22-Nov-2006
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540302742
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Mathematical Methods for Engineers and Scientists 1: Complex Analysis, Determinants and MatricesPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 22-Nov-2006
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540302742
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The topics of this set of student-oriented books are presented in a discursive style that is readable and easy to follow. 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 help students feel comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

Recenzijas

From the reviews:









"As the title suggests, this textbook in three volumes is mainly intended for students in natural sciences and engineering. This is of course a big advantage which could serve as a model for many textbooks in mathematics as well." (Jürgen Appell, Zentrablatt MATH, Vol. 1153, 2009)

Part I Complex Analysis
1 Complex Numbers
3(58)
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(3)
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
21(3)
1.4.1 Compound Interest
21(2)
1.4.2 The Limiting Process Representing e
23(1)
1.4.3 The Exponential Function ex
24(1)
1.5 Unification of Algebra and Geometry
24(4)
1.5.1 The Remarkable Euler Formula
24(1)
1.5.2 The Complex Plane
25(3)
1.6 Polar Form of Complex Numbers
28(18)
1.6.1 Powers and Roots of Complex Numbers
30(3)
1.6.2 Trigonometry and Complex Numbers
33(7)
1.6.3 Geometry and Complex Numbers
40(6)
1.7 Elementary Functions of Complex Variable
46(12)
1.7.1 Exponential and Trigonometric Functions of z
46(2)
1.7.2 Hyperbolic Functions of z
48(2)
1.7.3 Logarithm and General Power of z
50(5)
1.7.4 Inverse Trigonometric and Hyperbolic Functions
55(3)
Exercises
58(3)
2 Complex Functions
61(46)
2.1 Analytic Functions
61(20)
2.1.1 Complex Function as Mapping Operation
62(1)
2.1.2 Differentiation of a Complex Function
62(3)
2.1.3 Cauchy-Riemann Conditions
65(2)
2.1.4 Cauchy-Riemann Equations in Polar Coordinates
67(2)
2.1.5 Analytic Function as a Function of z Alone
69(5)
2.1.6 Analytic Function and Laplace's Equation
74(7)
2.2 Complex Integration
81(6)
2.2.1 Line Integral of a Complex Function
81(3)
2.2.2 Parametric Form of Complex Line Integral
84(3)
2.3 Cauchy's Integral Theorem
87(6)
2.3.1 Green's Lemma
87(2)
2.3.2 Cauchy-Goursat Theorem
89(1)
2.3.3 Fundamental Theorem of Calculus
90(3)
2.4 Consequences of Cauchy's Theorem
93(10)
2.4.1 Principle of Deformation of Contours
93(1)
2.4.2 The Cauchy Integral Formula
94(2)
2.4.3 Derivatives of Analytic Function
96(7)
Exercises
103(4)
3 Complex Series and Theory of Residues
107(66)
3.1 A Basic Geometric Series
107(1)
3.2 Taylor Series
108(9)
3.2.1 The Complex Taylor Series
108(1)
3.2.2 Convergence of Taylor Series
109(2)
3.2.3 Analytic Continuation
111(1)
3.2.4 Uniqueness of Taylor Series
112(5)
3.3 Laurent Series
117(9)
3.3.1 Uniqueness of Laurent Series
120(6)
3.4 Theory of Residues
126(15)
3.4.1 Zeros and Poles
126(2)
3.4.2 Definition of the Residue
128(1)
3.4.3 Methods of Finding Residues
129(4)
3.4.4 Cauchy's Residue Theorem
133(1)
3.4.5 Second Residue Theorem
134(7)
3.5 Evaluation of Real Integrals with Residues
141(24)
3.5.1 Integrals of Trigonometric Functions
141(3)
3.5.2 Improper Integrals I: Closing the Contour with a Semicircle at Infinity
144(3)
3.5.3 Fourier Integral and Jordan's Lemma
147(6)
3.5.4 Improper Integrals II: Closing the Contour with Rectangular and Pie-shaped Contour
153(5)
3.5.5 Integration Along a Branch Cut
158(2)
3.5.6 Principal Value and Indented Path Integrals
160(5)
Exercises
165(8)
Part II Determinants and Matrices
4 Determinants
173(40)
4.1 Systems of Linear Equations
173(6)
4.1.1 Solution of Two Linear Equations
173(2)
4.1.2 Properties of Second-Order Determinants
175(1)
4.1.3 Solution of Three Linear Equations
175(4)
4.2 General Definition of Determinants
179(9)
4.2.1 Notations
179(2)
4.2.2 Definition of a nth Order Determinant
181(2)
4.2.3 Minors, Cofactors
183(1)
4.2.4 Laplacian Development of Determinants by a Row (or a Column)
184(4)
4.3 Properties of Determinants
188(5)
4.4 Cramer's Rule
193(3)
4.4.1 Nonhomogeneous Systems
193(2)
4.4.2 Homogeneous Systems
195(1)
4.5 Block Diagonal Determinants
196(2)
4.6 Laplacian Developments by Complementary Minors
198(4)
4.7 Multiplication of Determinants of the Same Order
202(1)
4.8 Differentiation of Determinants
203(1)
4.9 Determinants in Geometry
204(4)
Exercises
208(5)
5 Matrix Algebra
213(42)
5.1 Matrix Notation
213(7)
5.1.1 Definition
213(1)
5.1.2 Some Special Matrices
214(2)
5.1.3 Matrix Equation
216(2)
5.1.4 Transpose of a Matrix
218(2)
5.2 Matrix Multiplication
220(13)
5.2.1 Product of Two Matrices
220(3)
5.2.2 Motivation of Matrix Multiplication
223(2)
5.2.3 Properties of Product Matrices
225(5)
5.2.4 Determinant of Matrix Product
230(2)
5.2.5 The Commutator
232(1)
5.3 Systems of Linear Equations
233(8)
5.3.1 Gauss Elimination Method
234(3)
5.3.2 Existence and Uniqueness of Solutions of Linear Systems
237(4)
5.4 Inverse Matrix
241(9)
5.4.1 Nonsingular Matrix
241(2)
5.4.2 Inverse Matrix by Cramer's Rule
243(3)
5.4.3 Inverse of Elementary Matrices
246(2)
5.4.4 Inverse Matrix by Gauss-Jordan Elimination
248(2)
Exercises
250(5)
6 Eigenvalue Problems of Matrices
255(58)
6.1 Eigenvalues and Eigenvectors
255(11)
6.1.1 Secular Equation
255(7)
6.1.2 Properties of Characteristic Polynomial
262(3)
6.1.3 Properties of Eigenvalues
265(1)
6.2 Some Terminology
266(5)
6.2.1 Hermitian Conjugation
267(1)
6.2.2 Orthogonality
268(1)
6.2.3 Gram-Schmidt Process
269(2)
6.3 Unitary Matrix and Orthogonal Matrix
271(7)
6.3.1 Unitary Matrix
271(1)
6.3.2 Properties of Unitary Matrix
272(1)
6.3.3 Orthogonal Matrix
273(1)
6.3.4 Independent Elements of an Orthogonal Matrix
274(1)
6.3.5 Orthogonal Transformation and Rotation Matrix
275(3)
6.4 Diagonalization
278(8)
6.4.1 Similarity Transformation
278(3)
6.4.2 Diagonalizing a Square Matrix
281(3)
6.4.3 Quadratic Forms
284(2)
6.5 Hermitian Matrix and Symmetric Matrix
286(12)
6.5.1 Definitions
286(1)
6.5.2 Eigenvalues of Hermitian Matrix
287(1)
6.5.3 Diagonalizing a Hermitian Matrix
288(8)
6.5.4 Simultaneous Diagonalization
296(2)
6.6 Normal Matrix
298(2)
6.7 Functions of a Matrix
300(9)
6.7.1 Polynomial Functions of a Matrix
300(1)
6.7.2 Evaluating Matrix Functions by Diagonalization
301(4)
6.7.3 The Cayley-Hamilton Theorem
305(4)
Exercises
309(4)
References 313(2)
Index 315
K.T. Tang received his B.S. in Engineering Physics and M.A. in Mathematics from 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 Boeing Company. Dr. Tang regards teaching as his calling, although his research accomplishments are also considerable. He authored/co-authored over 130 research papers in professional journals and a monograph "Asymptotic Methods in Quantum Mechanics". He lectured widely in Asia, Europe, and North America. He had been a long-term visiting scientist at Max-Planck-Institut in Göttingen. He is a recipient of a Distinguished U.S. Senior Scientist Award from Alexander von Humboldt Foundation and a Faculty Excellence Award from Pacific Lutheran University where he is Professor of Physics.
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