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Applied Complex Variables for Scientists and Engineers - Yue-Kuen Kwok

Applied Complex Variables for Scientists and Engineers

Paperback Published: 16th August 2010
ISBN: 9780521701389
Number Of Pages: 450

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This introduction to complex variable methods begins by carefully defining complex numbers and analytic functions, and proceeds to give accounts of complex integration, Taylor series, singularities, residues and mappings. Both algebraic and geometric tools are employed to provide the greatest understanding, with many diagrams illustrating the concepts introduced. The emphasis is laid on understanding the use of methods, rather than on rigorous proofs. Throughout the text, many of the important theoretical results in complex function theory are followed by relevant and vivid examples in physical sciences. This second edition now contains 350 stimulating exercises of high quality, with solutions given to many of them. Material has been updated and additional proofs on some of the important theorems in complex function theory are now included, e.g. the Weierstrass-Casorati theorem. The book is highly suitable for students wishing to learn the elements of complex analysis in an applied context.

'This text achieves a mixture of rigour and application that is not found in many books on complex variable theory. I think this is an advantage, allowing one to acquire the best of both mathematical precision alongside applications to physics and engineering ... This must surely appeal to a wide range of learning styles and ensure a greater understanding for anyone who chooses to read this text.' W. Joyce, Contemporary Physics
'... this book if acquired is bound to be consulted frequently on any time frame because of its breadth, versatility, completeness and mix of theory with applications.' W. Joyce, Contemporary Physics
'... clear and comprehensible ...' Bulletin of the Belgian Mathematical Society
'The book is just right for students learning the elements of function theory in an applied context.' Zeitschrift fur Angewandte Mathematik und Mechanik

Prefacep. ix
Complex Numbersp. 1
Complex numbers and their representationsp. 1
Algebraic properties of complex numbersp. 4
De Moivre's theoremp. 7
Geometric properties of complex numbersp. 13
nth roots of unityp. 16
Symmetry with respect to a circlep. 17
Some topological definitionsp. 23
Complex infinity and the Riemann spherep. 29
The Riemann sphere and stereographic projectionp. 30
Applications to electrical circuitsp. 33
Problemsp. 36
Analytic Functionsp. 46
Functions of a complex variablep. 46
Velocity of fluid flow emanating from a sourcep. 48
Mapping properties of complex functionsp. 50
Definitions of the exponential and trigonometric functionsp. 53
Limit and continuity of complex functionsp. 54
Limit of a complex functionp. 54
Continuity of a complex functionp. 58
Differentiation of complex functionsp. 61
Complex velocity and accelerationp. 63
Cauchy-Riemann relationsp. 64
Conjugate complex variablesp. 69
Analyticityp. 70
Harmonic functionsp. 74
Harmonic conjugatep. 75
Steady state temperature distributionp. 80
Poisson's equationp. 84
Problemsp. 85
Exponential, Logarithmic and Trigonometric Functionsp. 93
Exponential functionsp. 93
Definition from the first principlesp. 94
Mapping properties of the complex exponential functionp. 97
Trigonometric and hyperbolic functionsp. 97
Mapping properties of the complex sine functionp. 102
Logarithmic functionsp. 104
Heat sourcep. 106
Temperature distribution in the upper half-planep. 108
Inverse trigonometric and hyperbolic functionsp. 111
Generalized exponential, logarithmic, and power functionsp. 115
Branch points, branch cuts and Riemann surfacesp. 118
Joukowski mappingp. 123
Problemsp. 126
Complex Integrationp. 133
Formulations of complex integrationp. 133
Definite integral of a complex-valued function of a real variablep. 134
Complex integrals as line integralsp. 135
Cauchy integral theoremp. 142
Cauchy integral formula and its consequencesp. 151
Derivatives of contour integralsp. 153
Morera's theoremp. 157
Consequences of the Cauchy integral formulap. 158
Potential functions of conservative fieldsp. 162
Velocity potential and stream function of fluid flowsp. 162
Electrostatic fieldsp. 175
Gravitational fieldsp. 179
Problemsp. 183
Taylor and Laurent Seriesp. 194
Complex sequences and seriesp. 194
Convergence of complex sequencesp. 194
Infinite series of complex numbersp. 196
Convergence tests of complex seriesp. 197
Sequences and series of complex functionsp. 200
Convergence of series of complex functionsp. 201
Power seriesp. 206
Taylor seriesp. 215
Laurent seriesp. 221
Potential flow past an obstaclep. 230
Analytic continuationp. 233
Reflection principlep. 236
Problemsp. 238
Singularities and Calculus of Residuesp. 248
Classification of singular pointsp. 248
Residues and the Residue Theoremp. 255
Computational formulas for evaluating residuesp. 257
Evaluation of real integrals by residue calculusp. 268
Integrals of trigonometric functions over [0, 2]p. 268
Integrals of rational functionsp. 269
Integrals involving multi-valued functionsp. 271
Miscellaneous types of integralp. 275
Fourier transformsp. 278
Fourier inversion formulap. 279
Evaluation of Fourier integralsp. 285
Cauchy principal value of an improper integralp. 288
Hydrodynamics in potential fluid flowsp. 295
Blasius laws of hydrodynamic force and momentp. 295
Kutta-Joukowski's lifting force theoremp. 299
Problemsp. 300
Boundary Value Problems and Initial-Boundary Value Problemsp. 311
Integral formulas of harmonic functionsp. 312
Poisson integral formulap. 312
Schwarz integral formulap. 319
Neumann problemsp. 324
The Laplace transform and its inversionp. 326
Bromwich integralsp. 330
Initial-boundary value problemsp. 336
Heat conductionp. 337
Longitudinal oscillations of an elastic thin rodp. 341
Problemsp. 346
Conformal Mappings and Applicationsp. 358
Conformal mappingsp. 358
Invariance of the Laplace equationp. 364
Hodograph transformationsp. 372
Bilinear transformationsp. 375
Circle-preserving propertyp. 378
Symmetry-preserving propertyp. 381
Some special bilinear transformationsp. 390
Schwarz-Christoffel transformationsp. 399
Problemsp. 409
Answers to Problemsp. 419
Indexp. 434
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521701389
ISBN-10: 0521701384
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 450
Published: 16th August 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 2.2
Weight (kg): 0.72
Edition Number: 2
Edition Type: Revised