Complex Analysis with Applications in Science and Engineering

✍ Scribed by Harold Cohen

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 487
• Edition: 2nd
• Category: Library

✦ Synopsis

Complex Analysis with Applications in Science and Engineering weaves together theory and extensive applications in mathematics, physics and engineering. In this edition there are many new problems, revised sections, and an entirely new chapter on analytic continuation. This work will serve as a textbook for undergraduate and graduate students in the areas noted above.

Key Features of this Second Edition:

Excellent coverage of topics such as series, residues and the evaluation of integrals, multivalued functions, conformal mapping, dispersion relations and analytic continuation

Systematic and clear presentation with many diagrams to clarify discussion of the material

Numerous worked examples and a large number of assigned problems

✦ Table of Contents

0387730575......Page 1
Complex Analysis with Applications in Science and Engineering, Second Edition......Page 3
Contents......Page 7
Preface......Page 12
Examples......Page 16
Real Numbers......Page 21
Complex numbers......Page 22
Argand diagram......Page 23
Complex conjugation......Page 25
Addition and subtraction......Page 27
Multiplication......Page 29
Equality......Page 30
2.2 Cartesian, Trigonometric, and Polar Forms......Page 31
deMoivre’s theorem......Page 35
Exponential representations of trigonometric functions......Page 37
Hyperbolic functions......Page 38
Roots of 1......Page 39
Roots of –1......Page 42
2.4 Complex Numbers and AC Circuits......Page 43
DC circuits with resistors......Page 44
AC circuits with resistors, capacitors, and inductors......Page 46
Problems......Page 50
Derivatives and the Cauchy–Riemann conditions......Page 56
Analyticity......Page 59
Laplace’s equation for an analytic function......Page 63
Determination of an analytic function......Page 64
3.2 integrals of analytic function......Page 67
Removable and nonremovable singularities......Page 71
Pole singularities......Page 73
Cauchy’s residue theorem for one simple pole......Page 76
Cauchy’s residue theorem for more than one simple pole......Page 83
Cauchy’s residue theorem for high order poles......Page 86
Integrands with more than one high order pole......Page 88
Problems......Page 90
4.1 Taylor Series for an Analytic Function......Page 96
Convergence of the Taylor series......Page 98
4.2 Laurent Series for a Singular Function......Page 101
Laurent series for an analytic function......Page 104
Laurent series for a function with a pole of order M......Page 105
4.3 Radius of Convergence and the Cauchy Ratio......Page 107
4.4 Limits and Series......Page 111
Limit in the form ∞/∞......Page 114
Addition and subtraction......Page 115
Multiplication......Page 117
Division......Page 119
Contour integral of a function with a pole of order M......Page 123
Three methods for determining the residue......Page 126
Problems......Page 134
5.1 Integrals Along the Entire Real Axis......Page 145
Functions with inverse power asymptotic behavior......Page 148
Fourier exponential integrals......Page 150
Fourier sine and cosine integrals......Page 153
5.2 Integrals of Functions of Sine and Cosine......Page 155
5.3 Cauchy’s Principal Value Integral and theDirac δ Symbol......Page 157
Displacement of the pole and the Dirac δ symbol......Page 161
The Dirac δ symbol as a function......Page 167
5.4 Miscellaneous Integrals......Page 168
Problems......Page 176
Multivalued Functions, Branch Points, and Cuts......Page 183
n th power function......Page 184
General fractional root function......Page 185
Logarithm function......Page 187
6.2 Riemann Sheets, Branch Points, and Cuts......Page 188
Branch point......Page 189
Branch cut......Page 190
Construction of a physical model of a multisheeted complex plane......Page 191
Discontinuity across the cut......Page 192
6.3 Branch Structure......Page 193
Points on different sheets......Page 195
Cut structure for the fractional root function......Page 197
Cut structure for the logarithm function......Page 201
Functions with two branch points......Page 204
6.5 Evaluation of Integrals......Page 218
Specific examples......Page 220
Problems......Page 236
7.1 The Integrand Is Analytic......Page 242
7.2 The Integrand is Singular......Page 243
Movable singularity of the integrand......Page 244
Endpoint and pinch singularities......Page 245
Limits are analytic functions of z......Page 258
Integrals with limits that have singularities......Page 260
Problems......Page 261
8.1 Properties of a Mapping......Page 265
Into and onto mappings......Page 269
Invariant points......Page 270
Mapping of curves......Page 271
Tangent to a curve......Page 274
Conformal and isogonal mappings......Page 277
Inverse mapping......Page 279
Conformal mapping of a differential area......Page 283
Translation......Page 287
Magnification and rotation......Page 288
The bilinear transformation......Page 293
8.3 Schwarz–Christoffel Transformation......Page 298
Inverse of the SC mapping......Page 299
Mapping the real axis......Page 300
Mapping of points that are not on the real axis......Page 305
Closed form of the SC mapping......Page 307
Open and closed polygons......Page 314
Determining the SC mapping......Page 319
8.4 Applications......Page 327
Laplace’s equation......Page 328
Boundary conditions......Page 329
Applications of conformal mapping to problems in electrostatics......Page 334
Sources containing tables of mappings......Page 360
Problems......Page 361
Dispersion Relations......Page 383
9.1 Kramers-Kronig Dispersion Relations Over the Entire Real Axis......Page 384
Reflection symmetry around the imaginary axis......Page 388
9.3 Dispersion Relations for a Function With Branch Structure......Page 391
Subtracted Kramers–Kronig dispersion relations over the entire real axis......Page 397
Subtracted Kramers–Kronig dispersion relations over half the real axis......Page 399
Subtracted dispersion relations for a function with a branch point......Page 402
9.5 Dispersion Relations and a Representation of the Dirac δ Symbol......Page 406
Problems......Page 408
Analytic Continuation......Page 415
10.1 Analytic Continuation by Series......Page 417
Analytic continuation when the function represented by a series cannot be expressed in closed form......Page 422
Analytic continuation along an arc......Page 425
Analytic continuation along different arcs and the Schwarz reflection......Page 429
The gamma function......Page 434
Half integer factorials......Page 435
Approximation of z! for Re(z) positive and large......Page 436
Approximation of z! for lzl small......Page 439
Problem......Page 443
APPENDIX 1 AC CIRCUITS AND COMPLEX NUMBERS......Page 449
APPENDIX 2 DERIVATION OF GREEN’S THEOREM......Page 457
APPENDIX 3 DERIVATION OF THE GEOMETRIC SERIES......Page 462
APPENDIX 4 CAUCHY RATIO TEST FOR CONVERGENCE OFA SERIES......Page 463
APPENDIX 5
EVALUATION OF AN INTEGRAL......Page 466
APPENDIX 6
TRANSFORMATION OF LAPLACE’S EQUATION......Page 469
APPENDIX 7 TRANSFORMATION OF BOUNDARY CONDITIONS......Page 472
APPENDIX 8 THE BETA FUNCTION......Page 477
References......Page 479
Index......Page 481