Verfügbarkeit: Complex analysis with applications in science and engineering

Complex analysis with applications in science and engineering:

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Vorheriger Titel:	Cohen, Harold Fundamentals and applications of complex analysis
1. Verfasser:	Cohen, Harold (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York, NY Springer 2007
Ausgabe:	2. ed.
Schlagworte:	Analyse mathématique Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionentheorie Lehrbuch
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XXIII, 477 S. graph. Darst.
ISBN:	9780387730578

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Datensatz im Suchindex

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adam_text	Contents Preface ..........................................................................................................xv Example .......................................................................................................xix 1 INTRODUCTION ...................................................................................1 1.1 A Brief History .................................................................................1 Real numbers ....................................................................................1 Complex numbers .............................................................................2 Argand diagram .................................................................................3 2 COMPLEX NUMBERS .........................................................................5 2. ł Conjugation, Modulus, Argument and Arithmetic .........................................................................................5 Complex conjugation ........................................................................5 Modulus ............................................................................................7 Argument ..........................................................................................7 Addition and subtraction ...................................................................7 Multiplication ....................................................................................9 Division ...........................................................................................10 Equality ...........................................................................................10 2.2 Cartesian, Trigonometric and Polar Forms .....................................11 deMoivre s theorem ........................................................................15 Exponential representations of trigonometric functions .................17 Hyperbolic functions .......................................................................18 2.3 Roots of Unity ................................................................................19 Roots of 1........................................................................................19 Roots of -1......................................................................................22 Contents 2.4 Complex Numbers and AC Circuits ...............................................23 DC circuits with resistors ................................................................24 AC circuits with resistors, capacitors and inductors ......................26 Problems .................................................................................................30 COMPLEX VARIABLES .....................................................................37 3.1 Derivatives, Cauchy-Riemann Conditions and Analyticity .......................................................................................37 Derivatives and the Cauchy-Riemann conditions ...........................37 Analyticity .......................................................................................40 Laplace s equation for an analytic function ....................................44 Determination of an analytic function .............................................45 3.2 Integrals of Analytic Functions .......................................................48 3.3 Pole Singularities ............................................................................52 Removable and nonremovable singularities ...................................52 Pole singularities .............................................................................54 3.4 Cauchy s Residue Theorem ...........................................................57 Cauchy s residue theorem for one simple pole ...............................57 Cauchy s residue theorem for more than one simple pole ..............64 Cauchy s residue theorem for high order poles ..............................67 Integrands with more than one high order pole ...............................69 Problems .................................................................................................71 SERIES, LIMITS AND RESIDUES .....................................................77 4.1 Taylor Series for Analytic Functions .............................................77 Convergence of the Taylor series ...................................................79 4.2 Laurent Series for a Singular Function ..........................................82 Laurent series for an analytic function ...........................................85 Laurent series for a function with a pole of order M ......................86 4.3 Radius of Convergence and The Cauchy Ratio ..............................88 4.4 Limits and Series ............................................................................92 Limit in the form 0/0.......................................................................93 Limit in the form «>/«.....................................................................95 4.5 Arithmetic Combinations of Power Series ....................................96 Addition and subtraction .................................................................96 Multiplication .................................................................................98 Division .........................................................................................100 4.6 Residues .......................................................................................104 Contour integral of a function with a pole of order M .................104 Three methods for determining the residue ..................................107 Problems ..............................................................................................115 Contents xi 5 EVALUATION OF INTEGRALS.......................................................127 5.1 Integrals Along The Entire Real Axis ...........................................127 Functions with inverse power asymptotic behavior ......................130 Fourier exponential integrals ........................................................132 Fourier sine/cosine integrals .........................................................135 5.2 Integrals of Functions of sino and coso ......................................137 5.3 Cauchy s Principal Value Integral and The Dirac б Symbol .............................................................................139 Displacement of the pole and the Dirac б -symbol ........................ 143 The Dirac б -symbol as a function ................................................149 5.4 Miscellaneous Integrals ................................................................150 Problems ..............................................................................................158 6 MULTIVALUED FUNCTIONS, BRANCH POINTS AND CUTS ...................................................................................................165 6.1 Non-Integer Power, Logarithm Functions ....................................166 Hth power function ........................................................................166 General fractional root function ....................................................167 Irrational power function ...............................................................169 Logarithm function ........................................................................169 6.2 Riemann Sheets, Branch Points and Cuts .....................................170 Branch point ..................................................................................171 Branch cut .....................................................................................172 Construction of a physical model of a multisheeted complex plane ...........................................................................................173 Discontinuity across the cut ..........................................................174 6.3 Branch Structure ............................................................................175 Points on different sheets ..............................................................177 Cut structure for the fractional root function ................................179 Cut structure for the logarithm function ........................................183 6.4 Multiple Branch Points ..................................................................186 Functions with two branch points .................................................186 6.5 Evaluation of Integrals ..................................................................200 Specific examples ..........................................................................202 Problems ...............................................................................................218 7 SINGULARITIES OF FUNCTIONS DEFINED BY INTEGRALS .. 225 7.1 The Integrand is Analytic ..............................................................225 7.2 The Integrand is Singular ..............................................................226 Fixed singularity of the integrand .................................................227 Movable singularity of the integrand ............................................227 End point and pinch singularities ..................................................228 xii Contents 7.3 Limits of The Integral are Variable...............................................241 Limits are analytic functions of z ..................................................241 Integrals with limits that have singularities ...................................243 Problems ...............................................................................................244 8 CONFORMAL MAPPING ..................................................................249 8.1 Properties of a Mapping ................................................................249 Into and onto mappings ................................................................253 Invariant points .............................................................................254 Mapping of curves ........................................................................255 Tangent to a curve ........................................................................258 Conformai and isogonal mappings ...............................................261 Inverse mapping ............................................................................263 Conformai mapping of a differential area ....................................267 8.2 Linear and Bilinear Transformations ............................................271 The linear transformation .............................................................271 Translation ....................................................................................271 Magnification and rotation ...........................................................272 The bilinear transformation ..........................................................287 8.3 Schwarz-Christoffel Transformation ............................................282 Inverse of the SC mapping ...........................................................283 Mapping the real axis ....................................................................284 Mapping of points that are not on the real axis ............................289 Closed form of the SC mapping ...................................................291 Open and closed polygons ............................................................298 Determining the SC mapping .......................................................303 8.4 Applications ..................................................................................311 Laplace s equation ........................................................................312 Boundary conditions .....................................................................313 Applications of conformai mapping to problems 1 in electrostatics ..........................................................................318 Sources containing tables of mappings .........................................344 Problems ..............................................................................................345 9 DISPERSION RELATIONS ................................................................367 9.1 Kramers-Kronig Dispersion Relations Over The Entire Real Axis ....................................................................368 9.2 Kramers-Kronig Dispersion Relations Over Half The Real Axis ...............................................................................372 Reflection symmetry around the imaginary axis ..........................372 9.3 Dispersion Relations for a Function With Branch Structure ..................................................................375 Contents xiii Functions that are odd/even under reflection about the real axis ......................................................................................377 9.4 Subtracted Dispersion Relations ...................................................381 Subtracted Kramers-Kronig dispersion relations over the entire real axis ......................................................................................381 Subtracted Kramers-Kronig dispersion relations over half the real axis ......................................................................................383 Subtracted dispersion relations for a function with a branch point ...........................................................................................388 9.5 Dispersion Relations and a Representation of The Dirac б -Symbol ..................................................................... 390 Problems ...............................................................................................392 10 ANALYTIC CONTINUATION ..........................................................399 10.1 Analytic Continuation by Series .................................................401 Analytic continuation whe the function represented by a series cannot be expressed in closed form ..........................................406 Analytic continuation along an arc ..............................................409 Analytic continuation along different arcs and the Schwarz reflection ...................................................................................413 Natural boundaries ......................................................................416 10.2 Analytic Continuation of The Factorial ......................................418 The gamma function ....................................................................418 Half integers factorial ..................................................................419 Approximation of г! for Re(z) positive and large ........................420 Approximation of г! for z small ..............................................423 Problems ...............................................................................................427 APPENDIX 1 AC CIRCUITS AND COMPLEX NUMBERS .................433 Series AC circuit ................................................................436 Parallel AC circuit ..............................................................438 APPENDIX 2 DERIVATION OF GREEN S THEOREM .......................441 APPENDIX 3 DERIVATION OF THE GEOMETRIC SERIES ..............447 APPENDIX 4 CAUCHY RATIO TEST FOR CONVERGENCE OF A SERIES ............................................................................449 APPENDIX 5 EVALUATION OF AN INTEGRAL ................................453 APPENDIX 6 TRANSFORMATION OF LAPLACE^S EQUATION .....457 APPENDIX 7 TRANSFORMATION OF BOUNDARY CONDITIONS.461 APPENDIX 8 THE BETA FUNCTION ....................................................467 REFERENCES ...........................................................................................469 INDEX .......................................................................................................471
adam_txt	Contents Preface .xv Example .xix 1 INTRODUCTION .1 1.1 A Brief History .1 Real numbers .1 Complex numbers .2 Argand diagram .3 2 COMPLEX NUMBERS .5 2. ł Conjugation, Modulus, Argument and Arithmetic .5 Complex conjugation .5 Modulus .7 Argument .7 Addition and subtraction .7 Multiplication .9 Division .10 Equality .10 2.2 Cartesian, Trigonometric and Polar Forms .11 deMoivre's theorem .15 Exponential representations of trigonometric functions .17 Hyperbolic functions .18 2.3 Roots of Unity .19 Roots of 1.19 Roots of -1.22 Contents 2.4 Complex Numbers and AC Circuits .23 DC circuits with resistors .24 AC circuits with resistors, capacitors and inductors .26 Problems .30 COMPLEX VARIABLES .37 3.1 Derivatives, Cauchy-Riemann Conditions and Analyticity .37 Derivatives and the Cauchy-Riemann conditions .37 Analyticity .40 Laplace's equation for an analytic function .44 Determination of an analytic function .45 3.2 Integrals of Analytic Functions .48 3.3 Pole Singularities .52 Removable and nonremovable singularities .52 Pole singularities .54 3.4 Cauchy's Residue Theorem .57 Cauchy's residue theorem for one simple pole .57 Cauchy's residue theorem for more than one simple pole .64 Cauchy's residue theorem for high order poles .67 Integrands with more than one high order pole .69 Problems .71 SERIES, LIMITS AND RESIDUES .77 4.1 Taylor Series for Analytic Functions .77 Convergence of the Taylor series .79 4.2 Laurent Series for a Singular Function .82 Laurent series for an analytic function .85 Laurent series for a function with a pole of order M .86 4.3 Radius of Convergence and The Cauchy Ratio .88 4.4 Limits and Series .92 Limit in the form 0/0.93 Limit in the form «>/«.95 4.5 Arithmetic Combinations of Power Series .96 Addition and subtraction .96 Multiplication .98 Division .100 4.6 Residues .104 Contour integral of a function with a pole of order M .104 Three methods for determining the residue .107 Problems .115 Contents xi 5 EVALUATION OF INTEGRALS.127 5.1 Integrals Along The Entire Real Axis .127 Functions with inverse power asymptotic behavior .130 Fourier exponential integrals .132 Fourier sine/cosine integrals .135 5.2 Integrals of Functions of sino and coso .137 5.3 Cauchy's Principal Value Integral and The Dirac б Symbol .139 Displacement of the pole and the Dirac б -symbol . 143 The Dirac б -symbol as a function .149 5.4 Miscellaneous Integrals .150 Problems .158 6 MULTIVALUED FUNCTIONS, BRANCH POINTS AND CUTS .165 6.1 Non-Integer Power, Logarithm Functions .166 Hth power function .166 General fractional root function .167 Irrational power function .169 Logarithm function .169 6.2 Riemann Sheets, Branch Points and Cuts .170 Branch point .171 Branch cut .172 Construction of a physical model of a multisheeted complex plane .173 Discontinuity across the cut .174 6.3 Branch Structure .175 Points on different sheets .177 Cut structure for the fractional root function .179 Cut structure for the logarithm function .183 6.4 Multiple Branch Points .186 Functions with two branch points .186 6.5 Evaluation of Integrals .200 Specific examples .202 Problems .218 7 SINGULARITIES OF FUNCTIONS DEFINED BY INTEGRALS . 225 7.1 The Integrand is Analytic .225 7.2 The Integrand is Singular .226 Fixed singularity of the integrand .227 Movable singularity of the integrand .227 End point and pinch singularities .228 xii Contents 7.3 Limits of The Integral are Variable.241 Limits are analytic functions of z .241 Integrals with limits that have singularities .243 Problems .244 8 CONFORMAL MAPPING .249 8.1 Properties of a Mapping .249 Into and onto mappings .253 Invariant points .254 Mapping of curves .255 Tangent to a curve .258 Conformai and isogonal mappings .261 Inverse mapping .263 Conformai mapping of a differential area .267 8.2 Linear and Bilinear Transformations .271 The linear transformation .271 Translation .271 Magnification and rotation .272 The bilinear transformation .287 8.3 Schwarz-Christoffel Transformation .282 Inverse of the SC mapping .283 Mapping the real axis .284 Mapping of points that are not on the real axis .289 Closed form of the SC mapping .291 Open and closed polygons .298 Determining the SC mapping .303 8.4 Applications .311 Laplace's equation .312 Boundary conditions .313 Applications of conformai mapping to problems 1 in electrostatics .318 Sources containing tables of mappings .344 Problems .345 9 DISPERSION RELATIONS .367 9.1 Kramers-Kronig Dispersion Relations Over The Entire Real Axis .368 9.2 Kramers-Kronig Dispersion Relations Over Half The Real Axis .372 Reflection symmetry around the imaginary axis .372 9.3 Dispersion Relations for a Function With Branch Structure .375 Contents xiii Functions that are odd/even under reflection about the real axis .377 9.4 Subtracted Dispersion Relations .381 Subtracted Kramers-Kronig dispersion relations over the entire real axis .381 Subtracted Kramers-Kronig dispersion relations over half the real axis .383 Subtracted dispersion relations for a function with a branch point .388 9.5 Dispersion Relations and a Representation of The Dirac б -Symbol . 390 Problems .392 10 ANALYTIC CONTINUATION .399 10.1 Analytic Continuation by Series .401 Analytic continuation whe the function represented by a series cannot be expressed in closed form .406 Analytic continuation along an arc .409 Analytic continuation along different arcs and the Schwarz reflection .413 Natural boundaries .416 10.2 Analytic Continuation of The Factorial .418 The gamma function .418 Half integers factorial .419 Approximation of г! for Re(z) positive and large .420 Approximation of г! for \z\ small .423 Problems .427 APPENDIX 1 AC CIRCUITS AND COMPLEX NUMBERS .433 Series AC circuit .436 Parallel AC circuit .438 APPENDIX 2 DERIVATION OF GREEN'S THEOREM .441 APPENDIX 3 DERIVATION OF THE GEOMETRIC SERIES .447 APPENDIX 4 CAUCHY RATIO TEST FOR CONVERGENCE OF A SERIES .449 APPENDIX 5 EVALUATION OF AN INTEGRAL .453 APPENDIX 6 TRANSFORMATION OF LAPLACE^S EQUATION .457 APPENDIX 7 TRANSFORMATION OF BOUNDARY CONDITIONS.461 APPENDIX 8 THE BETA FUNCTION .467 REFERENCES .469 INDEX .471
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spelling	Cohen, Harold Verfasser aut Complex analysis with applications in science and engineering Harold Cohen 2. ed. New York, NY Springer 2007 XXIII, 477 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Analyse mathématique Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 1. Auflage Cohen, Harold Fundamentals and applications of complex analysis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016595881&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Cohen, Harold Complex analysis with applications in science and engineering Analyse mathématique Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd
subject_GND	(DE-588)4018935-1 (DE-588)4123623-3
title	Complex analysis with applications in science and engineering
title_auth	Complex analysis with applications in science and engineering
title_exact_search	Complex analysis with applications in science and engineering
title_exact_search_txtP	Complex analysis with applications in science and engineering
title_full	Complex analysis with applications in science and engineering Harold Cohen
title_fullStr	Complex analysis with applications in science and engineering Harold Cohen
title_full_unstemmed	Complex analysis with applications in science and engineering Harold Cohen
title_old	Cohen, Harold Fundamentals and applications of complex analysis
title_short	Complex analysis with applications in science and engineering
title_sort	complex analysis with applications in science and engineering
topic	Analyse mathématique Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd
topic_facet	Analyse mathématique Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionentheorie Lehrbuch
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