دانلود کتاب مسیرهای ریاضی به دنیای فیزیکی

1. Complex Functions 1
1.1 Complex Numbers . . . . . . . . . . . . . . . . . . 1
1.2 Functions of One Complex Variable . . . . . . . . . 14
1.3 Derivatives of Complex Functions . . . . . . . . . . 25
1.4 Analytic Functions . . . . . . . . . . . . . . . . . . 29
1.5 Multivalued Functions . . . . . . . . . . . . . . . . 34
1.6 Complex Potential Functions . . . . . . . . . . . . 43
2. Path Integrals 49
2.1 Complex Line Integrals . . . . . . . . . . . . . . . . 49
2.2 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . 52
2.3 Cauchy’s Integral Formula . . . . . . . . . . . . . . 60
2.4 Integration of Multivalued Functions . . . . . . . . 68
2.5 Elliptic Integrals . . . . . . . . . . . . . . . . . . . 71
3. Series Expansions 83
3.1 Convergence of Series . . . . . . . . . . . . . . . . . 83
3.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . 93
3.3 Laurent Series . . . . . . . . . . . . . . . . . . . . . 101
3.4 Types of Singularities . . . . . . . . . . . . . . . . 108
3.5 Singular Planar Fields . . . . . . . . . . . . . . . . 115
4. Calculus of Residues 121
4.1 Residue Theorem . . . . . . . . . . . . . . . . . . . 121
4.2 Integration of Real Functions . . . . . . . . . . . . 126
4.3 Special Integrals . . . . . . . . . . . . . . . . . . . 140
4.4 Series Summation . . . . . . . . . . . . . . . . . . . 154
5. Analytic Theory 165
5.1 Analytic Continuation . . . . . . . . . . . . . . . . 165
5.2 Analytic Continuation Functions . . . . . . . . . . 173
5.3 Logarithmic Integrals . . . . . . . . . . . . . . . . . 190
5.4 Decomposition of Meromorphic Functions . . . . . 195
5.5 Decomposition of Entire Functions . . . . . . . . . 204
6. Conformal Mappings 211
6.1 Conformal Transformation . . . . . . . . . . . . . . 211
6.2 Elementary Function Transformations . . . . . . . 218
6.3 Joukowsky Transformation . . . . . . . . . . . . . . 233
6.4 Polygon Mapping . . . . . . . . . . . . . . . . . . . 239
6.5 Conformal Self-Maps . . . . . . . . . . . . . . . . . 249
7. Fourier Analysis 261
7.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . 261
7.2 Fourier Transform . . . . . . . . . . . . . . . . . . 275
7.3 Convolution Theorem . . . . . . . . . . . . . . . . 291
7.4 Poisson Summation Formula . . . . . . . . . . . . . 299
8. Function Transformations 305
8.1 Laplace Transform . . . . . . . . . . . . . . . . . . 305
8.2 Laplace Inverse Transform . . . . . . . . . . . . . . 312
8.3 Some Applications . . . . . . . . . . . . . . . . . . 318
8.4 Z-Transform . . . . . . . . . . . . . . . . . . . . . . 327
9. General Solutions of Differential Equations 339
9.1 Constant Coefficient Ordinary Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . 339
9.2 Variable Coefficient Ordinary Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . 346
9.3 Constant Coefficient Partial Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . 365
9.4 Nonlinear Equations . . . . . . . . . . . . . . . . . 372
10. Differential Equations and Boundary Conditions 389
10.1 Mathematical Physics Equations . . . . . . . . . . 389
10.2 Boundary Value Problems . . . . . . . . . . . . . . 400
10.3 D’ Alembert’s Formula . . . . . . . . . . . . . . . . 406
10.4 Orthogonal Curvilinear Coordinates . . . . . . . . 413
10.5 Classification of Partial Differential Equations . . . 421
11. Separation of Variables Methods 427
11.1 Homogeneous Boundary Problems . . . . . . . . . 427
11.2 Non-Homogeneous Boundary Problems . . . . . . . 444
11.3 Periodic Boundary Problems . . . . . . . . . . . . 450
11.4 Joining Problems . . . . . . . . . . . . . . . . . . . 460
12. Integral Transform Methods 469
12.1 Generalized Functions . . . . . . . . . . . . . . . . 469
12.2 Fourier TransformMethod . . . . . . . . . . . . . . 480
12.3 Laplace TransformMethod . . . . . . . . . . . . . 494
13. Spherical Harmonics 509
13.1 Legendre Equation . . . . . . . . . . . . . . . . . . 509
13.2 Associated Legendre Equation . . . . . . . . . . . . 529
13.3 Spherical Harmonic Functions . . . . . . . . . . . . 533
14. Eigenfunction Theory 547
14.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . 547
14.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 556
14.3 Sturm–Liouville Systems . . . . . . . . . . . . . . . 565
14.4 Eigenvalue Theory . . . . . . . . . . . . . . . . . . 574
14.5 Classical Orthogonal Polynomials . . . . . . . . . . 592
15. Special Functions 603
15.1 Bessel Functions . . . . . . . . . . . . . . . . . . . 603
15.2 Modified Bessel Function . . . . . . . . . . . . . . . 618
15.3 Spherical Bessel Functions . . . . . . . . . . . . . . 625
15.4 Classification of Special Functions . . . . . . . . . . 635
15.5 Confluent Hypergeometric Function . . . . . . . . . 650
16. Green’s Functions 657
16.1 Fundamental Theory . . . . . . . . . . . . . . . . . 657
16.2 Potential Equations . . . . . . . . . . . . . . . . . . 670
16.3 Some Applications . . . . . . . . . . . . . . . . . . 679
16.4 Development Equation . . . . . . . . . . . . . . . . 682
16.5 PerturbationMethod . . . . . . . . . . . . . . . . . 688
17. Calculus of Variations 697
17.1 Functionals and Extremals . . . . . . . . . . . . . . 697
17.2 Euler–Lagrange Equation . . . . . . . . . . . . . . 703
17.3 Constrained Systems . . . . . . . . . . . . . . . . . 713
17.4 Mathematical Principles of Physics . . . . . . . . . 720
17.5 Boundary Value Problems . . . . . . . . . . . . . . 733
17.6 Rayleigh–Ritz Approximation . . . . . . . . . . . . 738
Appendix 745
Name Index 755
Subject Index 757