Complex analysis with applications to flows and fields:
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| Format: | Book |
| Language: | English |
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Boca Raton, Fla. [u.a.]
CRC Press
2011
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| Series: | Mathematics and physics for science and technology
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XL, 989 S. Ill., graph. Darst. |
| ISBN: | 1420071181 9781420071184 |
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| 245 | 1 | 0 | |a Complex analysis with applications to flows and fields |c L. M. B. C. Campos |
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| adam_text | IMAGE 1
MATHEMATICS AND PHYSICS FOR SCIENCE AND TECHNOLOGY
COMPLEX ANALYSIS WITH APPLICATIONS TO FLOWS AND FIELDS
L.M.B.C. CAMPOS DIRECTOR OFTHE CENTER FOR AERONAUTICAL
AND SPACE SCIENCE AND TECHNOLOGYLISBONTECHNICAL UNIVERSITY
CRC PRESS TAYLOR& FRANCIS GROUP BOCA RATON LONDON NEW YORK
CRC PRESS IS AN IMPRINT OF THE TAYLOR & FRANCIS GROUP, AN INFORMA
BUSINESS
IMAGE 2
CONTENTS
LIST OF TABLES, NOTES, DIAGRAMS, CLASSIFICATIONS, AND LISTS XVII
SERIES PREFACE XXI
PREFACE XXV
ABOUT THE AUTHOR XXVII
ACKNOWLEDGMENTS XXIX
MATHEMATICAL SYMBOLS XXXI
PHYSICAL QUANTITIES XXXVII
PART 1 COMPLEX DOMAIN: CIRCUITS AND STABILITY 1
1 COMPLEX NUMBERS AND QUATERNIONS 3
1.1 PEANO (1889, 1891) POSTULATES FOR NATURAL NUMBERS 3
1.2 IRRATIONAL NUMBERS (PYTHAGORAS, VI B.C.) AND DEDEKIND (1858) SECTION
. . 5 1.3 CARTESIAN PARTS: REAL AND IMAGINARY (ARGAND, 1806; DESCARTES,
1637A; GAUSS, 1797) 6
1.4 POLAR COORDINATES: MODULUS AND ARGUMENT 6
1.5 MOIVRE S FORMULA, ORIGIN AND INFINITY 7
1.6 CONJUGATE AND REFLECTION ON THE ORIGIN AND AXIS 8
1.7 POWER WITH INTEGRAL EXPONENT AND LOGARITHM 9
1.8 REAL, IMAGINARY, AND COMPLEX EXPONENTIAL 9
1.9 NONCOMMUTATIVE PRODUCT OF QUATERNIONS (HAMILTON, 1843) 10
2 STABILITY OF AN EQUILIBRIUM POSITION 13
2.1 TRAJECTORY FOLLOWING A PERTURBATION OF EQUILIBRIUM 13
2.2 OSCILLATORY MOTION WITH CONSTANT AMPLITUDE 14
2.3 ATTENUATION OR AMPLIFICATION AND STABILITY OR INSTABILITY 15
2.4 DAMPED OSCILLATION OR OVERSTABLE GROWTH 15
2.5 GENERAL RELATIONS FOR AMPLITUDES AND PHASES 16
2.6 PREDOMINANTLY OR WEAKLY OSCILLATORY MOTION 16
2.7 FREQUENCY AND ATTENUATION/AMPLIFICATION FACTOR 17
2.8 DIFFERENTIAL EQUATION AND STABILITY CRITERIA 18
2.9 INITIAL CONDITIONS FOR HARMONIC OSCILLATOR 19
3 ADDITION, PRODUCT, AND INVERSES 23
3.1 COMPLEX ADDITION AND RULE OF THE PARALLELOGRAM 23
3.2 MODULUS, ARGUMENT, AND TRIANGULAR EQUALITIES (PYTHAGORAS, VI B.C.)
... 24 3.3 COMPLEX PRODUCT, HOMOTHETY, AND ROTATION 25
3.4 MEANING OF THE IMAGINARY SYMBOL I 26
3.5 CONJUGATE OF THE SUM, PRODUCT, AND INVERSION 27
3.6 COMPLEX REPRESENTATION OF REAL QUANTITIES 27
3.7 TRIGONOMETRIC ADDITION AND MULTIPLICATION FORMULAS 28
3.8 CONJUGATE COMPLEX AND TRIANGULAR INEQUALITIES 30
3.9 GENERALIZED SCHWARTZ (1890) OR POLYGONAL INEQUALITY 30
VII
IMAGE 3
VIII CONTENTS
4 IMPEDANCE OF ASSOCIATIONS OF CIRCUITS 33
4.1 INERTIA, FRICTION, AND ELASTIC FORCES 33
4.2 FREE AND FORCED MOTION OF CIRCUIT 34
4.3 ELECTRICAL INDUCTION, RESISTANCE, AND CAPACITY 35
4.4 DECOMPOSITION OF IMPEDANCE INTO INDUCTANCE AND REACTANCE 36
4.5 ACTIVITY IN TERMS OF THE VELOCITY, FORCE, AND IMPEDANCE 36
4.6 MECHANICAL CIRCUITS IN PARALLEL OR SERIES 38
4.7 ELECTROMECHANICAL ANALOGY AND CONTRASTING LAWS 39
4.8 COMPARISON OF TWO CIRCUITS IN PARALLEL AND IN SERIES 40
4.9 HYBRID ASSOCIATIONS OF THREE CIRCUITS 40
5 POWER, ROOT, AND LOGARITHM 43
5.1 RA-TH POWER AS THE (N - 1)-TIMES ITERATED PRODUCT 43
5.2 DISCRETE SET OF POINTS ON LOGARITHMIC SPIRAL 44
5.3 INVERSION OF THE POWER: ROOTS OF ORDER N 45
5.4 REGULAR POLYGON CONTAINED IN A CIRCLE 45
5.5 MULTIPLE SUMS OF SINES OR COSINES OF EQUAL ANGLES 47
5.6 SINGLE-, MULTI-, AND MANY-VALUED FUNCTIONS 48
5.7 POWER WITH COMPLEX BASE AND EXPONENT 49
5.8 LIMITING BEHAVIOR AT THE ORIGIN AND INFINITY 49
5.9 VANISHING AND DIVERGENCE ON ALTERNATE SECTORS 50
6 ELECTRON IN AN ELECTROMAGNETIC FIELD 53
6.1 ELECTROMAGNETIC OR LAPLACE-LORENTZ FORCE 53
6.2 UNIFORM FIELDS AND LARMOR (1897) FREQUENCY 54
6.3 LONGITUDINAL TRANSLATION AND TRANSVERSE ROTATION 54
6.4 COMPONENTS OF THE VELOCITY AND TRAJECTORY OF THE PARTICLE 55
6.5 LINEAR, CIRCULAR, AND HELICAL MOTION 55
6.6 LINEAR ACCELERATION AND CYCLOID IN THE PLANE 57
6.7 ELONGATED HELIX AND PLANE TROCHOID 59
6.8 OVAL AND CONICAL HELICES AND MAGNETIC FOCUSING 60
6.9 SEPARATION OF ISOTOPES IN A MASS SPECTROGRAPH 62
7 MULTIVALUED FUNCTIONS, BRANCH-POINTS, AND BRANCH-CUTS 65 7.1 RIEMANN
(1857) SURFACE OF A MULTIVALUED FUNCTION 65
7.2 DENUMERABLE INFINITY OF CONNECTED SHEETS 67
7.3 PRINCIPAL BRANCH IN THE CUT-PLANE 69
7.4 JUMP DISCONTINUITY ACROSS A BRANCH-CUT 70
7.5 SEMIINFINITE CUT JOINING A BRANCH-POINT TO INFINITY 70
7.6 INFINITE DERIVATIVE OF A FUNCTION AT A BRANCH-POINT 71
7.7 THEOREM AND METHOD FOR THE IDENTIFICATION OF BRANCH-POINTS 72
7.8 ELEMENTARY FUNCTIONS WITH TWO BRANCH-POINTS 73
7.9 FUNCTIONS WITH SEVERAL BRANCH-POINTS AND BRANCH-CUTS 77
8 MOTION OF A PENDULUM AND A SHIP 81
8.1 STABILITY OF A SUSPENDED OR INVERTED PENDULUM 81
8.2 MOTION OF THE PENDULUM AND FORCE ALONG THE ROD 83
8.3 METACENTRIC DISTANCE AND ROLLING TORQUE 84
8.4 LENGTH OF THE PENDULUM EQUIVALENT TO THE SHIP 84
8.5 KINETIC, POTENTIAL, AND TOTAL ENERGY 85
8.6 LINEARIZATION IN THE VICINITY OF THE EQUILIBRIUM POSITION 86
IMAGE 4
CONTENTS IX
8.7 BRANCH-POINTS AS BOUNDARIES OF THE ANGULAR MOTION 87
8.8 OSCILATION ABOUT A POSITION OF STABLE EQUILIBRIUM (GALILEO, 1583) 88
8.9 DIVERGENCE AWAY FROM THE POSITION OF UNSTABLE EQUILIBRIUM 89
9 STEREOGRAPHIC PROJECTION AND GENUS OF A SURFACE 91
9.1 INJECTIVE, SURJECTIVE, AND BIJECTIVE MAPPINGS 91
9.2 UNIT SPHERE AND THE COMPLEX PLANE 93
9.3 DIRECT AND INVERSE STEREOGRAPHIC TRANSFORMATION 94
9.4 MAPPING OF A CIRCLE ONTO A STRAIGHT LINE 95
9.5 PROJECTION OF A CIRCLE INTO ANOTHER CIRCLE 95
9.6 ISOMORPHISM OF CIRCLE AND REAL LINE 96
9.7 CONTINUOUS DEFORMATION AND TOPOLOGICAL SPHERE 97
9.8 SPHERE WITH ONE HANDLE AND TOROIDAL TOPOLOGY 98
9.9 TORUS WITH HOLES OR SPHERE WITH HANDLES 99
10 EXAMPLES 10.1 TO 10.20 103
PART 2 INTEGRALS AND RESIDUES: FLOWS AND GRAVITY 121
11 DIFFERENTIATION AND HOLOMORPHIC FUNCTIONS 123
11.1 FUNCTION, NEIGHBORHOOD, LIMIT, AND UNIFORMITY 123
11.2 CONTINUITY, INCREMENTAL RATIO, AND DERIVATE (NEWTON, 1670;
LEIBNITZ, 1684) 124
11.3 HOLOMORPHIC FUNCTION AND CONTINUOUS DERIVATIVES 125
11.4 CAUCHY (1821)-RIEMANN (1851) CONDITIONS IN CARTESIAN AND POLAR
COORDINATES 128
11.5 FORMULAS FOR THE DERIVATIVE AND ITS MODULUS AND ARGUMENT 128
11.6 CARTESIAN AND POLAR LAPLACE (1825) EQUATION 129
11.7 GRADIENT, DIVERGENCE, CURL, AND LAPLACIAN 130
11.8 FAMILIES OF PLANE ORTHOGONAL CURVES 131
11.9 ORTHOGONAL PLANE CURVILINEAR COORDINATES 132
12 POTENTIAL FLOW AND MULTIPOLES 135
12.1 CIRCULATION, POTENTIAL, CURL, AND VORTICITY 135
12.2 FLOW RATE, STREAM FUNCTION, DIVERGENCE, AND DILATATION (LAGRANGE,
1781; RANKINE, 1864) 137
12.3 COMPLEX POTENTIAL AND CONJUGATE VELOCITY 138
12.4 IRROTATIONAL FLOW DUE TO A SOURCE OR SINK 139
12.5 INCOMPRESSIBLE FLOW DUE TO A VORTEX 141
12.6 SUPERPOSITION AS A MONOPOLE AND SPIRAL FLOW 142
12.7 DIPOLE AS THE LIMIT OF TWO OPPOSING MONOPOLES 142
12.8 QUADRUPOLE MOMENT AND RULE OF DIFFERENTIATION 145
12.9 ARBITRARY MULTIPOLE AND DIRECTIVITY LOBES 147
13 PRIMITIVE AND CONTOUR INTEGRALS 151
13.1 EXISTENCE AND PROPERTIES OF THE PRIMITIVE OF A FUNCTION 151
13.2 RIEMANN INTEGRAL OF A COMPLEX FUNCTION (CAUCHY, 1825; RIEMANN,
1851) 152
13.3 RECTIFIABLE CURVES AND BOUNDED FUNCTIONS 154
13.4 PARAMETRIC LIMITS AND UNIFORM CONTINUITY 155
13.5 COMPLEX LOOP AND CONTOUR INTEGRALS 156
IMAGE 5
X CONTENTS
13.6 RECIPROCAL THEOREMS OF CAUCHY (1825) AND MORERA (1886)~OSGOOD
(1896) 157
13.7 INTEGRATION BY PARTS AND CHAIN RULE (LEIBNITZ, 1864) 158
13.8 DERIVATION OF AN INTEGRAL WITH REGARD TO A PARAMETER 159
13.9 PARAMETRIC INTEGRAL WITH VARIABLE END-POINTS 160
14 PRESSURE AND CORNER FLOWS 163
14.1 MASS CONSERVATION AND EQUATION OF CONTINUITY 163
14.2 INVISCID MOMENTUM EQUATION (EULER, 1752, 1759) 165
14.3 ADIABATIC CONDITION AND EQUATION OF STATE 166
14.4 HOMENTROPIC FLOW AND CONSERVATION OF CIRCULATION (HELMHOLTZ, 1858;
KELVIN, 1869) 167
14.5 HYDROSTATIC, DYNAMIC, AND STAGNATION PRESSURES (TORRICELLI, 1643;
BERNOULLI, 1738) 168
14.6 COMPRESSIBILITY EFFECTS AND THE PITOT TUBE (1732) 169
14.7 VENTURI TUBE (HERSCHEL, 1887) AND VARIABLE-AREA DUCT 173
14.8 CORNER FLOWS AND MULTIPOLES AT INFINITY 177
14.9 STREAM PAST A WEDGE AND SHARP EDGE 181
15 LOOP INTEGRALS AND POLES 187
15.1 CAUCHY (1821) FIRST THEOREM ON INTEGRALS 187
15.2 DOUBLY-CONNECTED REGION AND SHRINKING OF A LOOP 188
15.3 SECOND CAUCHY (1821) THEOREM: VALUE OF THE FUNCTION 189
15.4 THIRD CAUCHY (1821) THEOREM: ALL THE DERIVATES 190
15.5 INCLUSION OR EXCLUSION OF SINGULARITIES ON THE BOUNDARY 191
15.6 HOLOMORPHIC FUNCTION IN A MULTIPLY-CONNECTED REGION 193
15.7 RESIDUE OF A FUNCTION AT A SIMPLE POLE 194
15.8 MULTIPLE POLE OR POLE OF ORDER N 196
15.9 LOOP INTEGRAL WITH POLES IN THE INTERIOR AND ON THE BOUNDARY 198
16 IMAGES ON PLANE WALLS 201
16.1 IDENTICAL IMAGE ON A RIGID WALL (RANKINE, 1864) 201
16.2 IMAGE VORTEX WITH OPPOSITE CIRCULATION 206
16.3 EFFECT OF WALL ON MONOPOLE OR SPIRAL SOURCE 209
16.4 FAR-FIELD OF MULTIPOLE NEAR A HARD WALL 210
16.5 MONOPOLE IN A HARD-WALLED RECTANGULAR CORNER 211
16.6 TRAJECTORIES OF A VORTEX OR SOURCE/SINK IN A CORNER 212
16.7 FLOW AND FORCES FOR A MONOPOLE IN A RECTANGULAR CORNER (GROBLI,
1877; GREENHILL, 1878) 215
16.8 MULTIPLE IDENTICAL SOURCE/SINK IMAGES 220
16.9 ALTERNATING VORTICES IN A RIGID CORNER 221
17 IMPROPER INTEGRALS AND PRINCIPAL VALUE 225
17.1 IMPROPER UNI(BI)LATERAL INTEGRALS OF THE THREE KINDS 225
17.2 TRANSFORMATION OF A STRAIGHT SEGMENT INTO A CIRCLE 226
17.3 CLOSING A STRAIGHT LINE BY A HALF-CIRCLE 228
17.4 CONNECTING THE REAL AXIS IN THE UPPER/LOWER HALF-PLANE 229
17.5 INTEGRALS WITH AN OSCILLATING FACTOR (JORDAN, 1894) 231
17.6 THE LOCALIZATION LEMMA FOR HOLOMORPHIC FUNCTIONS 234
17.7 SURROUNDING A SEMIINFMITE BRANCH-CUT 235
IMAGE 6
CONTENTS XI
17.8 BRANCH-POINT WITHIN THE PATH OF INTEGRATION 238
17.9 CAUCHY (1821) PRINCIPAL VALUE OF AN INTEGRAL 239
18 MASS AND THE GRAVITY FIELD 243
18.1 IRROTATIONAL FLOW DUE TO SOURCES OR SINKS 243
18.2 INCOMPRESSIBLE FLOW DUE TO A VORTICITY DISTRIBUTION 245
18.3 GRAVITY FIELD AND GRAVITATIONAL CONSTANT 246
18.4 LINE, SURFACE, AND VOLUME MASS DISTRIBUTIONS 247
18.5 GRAVITY FORCE OF ATTRACTION (NEWTON, 1687) 248
18.6 GRAVITY FIELD OF A HOMOGENEOUS SLAB 249
18.7 GRAVITY FIELD INSIDE AND OUTSIDE THE MASS 251
18.8 FIELD DUE TO A DISTRIBUTION OF INFINITE EXTENT 253
18.9 MULTIPOLAR REPRESENTATION OF THE GRAVITY FIELD 256
19 CAUCHY CONDITIONS AND INFINITESIMALS 261
19.1 CALCULATION OF RIEMANN INTEGRALS USING THE DEFINITION 261
19.2 MEAN VALUE THEOREM AND BOUNDS 262
19.3 DIVISION INTO INTERNAL AND BOUNDARY REGIONS 262
19.4 FUNCTION HOLOMORPHIC IN THE INTERIOR AND ON THE BOUNDARY 264
19.5 UNIFORM CONTINUITY ON THE BOUNDARY (GOURSAT, 1900) 265
19.6 ISOLATED IGNORABLE SINGULARITIES ON THE BOUNDARY (LITTLEWOOD, 1944)
265
19.7 INFINITESIMALS OF THE SAME OR HIGHER ORDER 266
19.8 ZERO OF ORDER N AND L HOSPITAL S (1696; BERNOULLI, 1691).RULE 267
19.9 CALCULATION OF THE RESIDUES OF RATIOS OF FUNCTIONS 269
20 EXAMPLES 20.1 TO 20.20 273
PART 3 POWER SERIES: ELECTRICITY AND MAGNETISM 295
21 CONVERGENCE OF AND OPERATIONS ON SERIES 297
21.1 CONVERGENT, DIVERGENT, AND OSCILLATING SERIES 297
21.2 ASSOCIATION OF TERMS AND SUM OF A SERIES 300
21.3 ABSOLUTE AND CONDITIONAL CONVERGENCE (DIRICHLET, 1837) 301
21.4 PERMUTATION OF TERMS AND PRODUCT OF SERIES (CAUCHY, 1821) 303
21.5 UNIFORM CONVERGENCE AND SERIES OF FUNCTIONS 304
21.6 LIMIT, DIFFERENTIATION, AND INTEGRATION TERM-BY-TERM 308
21.7 TOTAL CONVERGENCE AND WEIERSTRASS M-TEST (1876) 310
21.8 GEOMETRIC, LOGARITHMIC, AND INVERSE-POWER SERIES 311
21.9 CONVERGENCE INSIDE, OUTSIDE, AND ON THE UNIT CIRCLE 313
22 MULTIPLE REFLECTIONS IN A LENS 317
22.1 PERIOD, FREQUENCY, WAVELENGTH, AND WAVEVECTOR 317
22.2 REFLECTION, TRANSMISSION, AND INACCESSIBLE REGIONS (SNELL, 1626;
DESCARTES, 1637B; FRESNEL, 1823) 319
22.3 WAVE SCATTERING AND FASTEST PATH (FERMAT, 1657) 321
22.4 REFLECTION AND TRANSMISSION OF ACOUSTIC WAVES 323
22.5 ADSORPTION AT AN INTERFACE AND INTERNAL ABSORPTION 326
22.6 MULTIPLE REFLECTIONS BETWEEN PARALLEL INTERFACES 328
22.7 TOTAL REFLECTION, TRANSMISSION, AND DAMPING COEFFICIENTS 329
22.8 MULTIPLE MEDIA, TRANSPARENCY, AND OPAQUENESS 330
22.9 CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE (BRAGG, 1912) 331
IMAGE 7
XU CONTENTS
23 ANALYTIC SERIES OF ASCENDING POWERS 335
23.1 HARMONIC FUNCTION AND MEAN VALUE ON A CIRCLE 335
23.2 LEMMAS OF CONSTANCY AND MAXIMUM MODULUS 336
23.3 MONOTONIC CHAIN OF REGIONS AND LOOPS 337
23.4 GEOMETRIC SERIES OF HOLOMORPHIC FUNCTIONS 338
23.5 REGIONS OF ABSOLUTE AND UNIFORM CONVERGENCE 339
23.6 LAGRANGE (1770)-BURMANN (1799) SERIES AND IMPLICIT DERIVATION 340
23.7 TAYLOR (1715) AND STIRLING (1717)-MACLAURIN (1742) THEOREMS 341
23.8 IMPLICIT DERIVATIVES AND MEAN-VALUE THEOREM 342
23.9 DARBOUX EXPANSION (1876) AND LAGRANGE/CAUCHY REMAINDERS 343
24 ELECTROSTATICS, CHARGES, AND DIELECTRICS 353
24.1 ELECTRIC FIELD, DISPLACEMENT, AND POLARIZATION (MAXWELL, 1873) 353
24.2 DIELECTRIC PERMITTIVITY AND ELECTRIC SUSCEPTIBILITY 354
24.3 POTENTIAL DUE TO CHARGES AND ELECTRIC FORCE (COULOMB, 1785) 355
24.4 MULTIPOLE NEAR INSULATING OR CONDUCTING WALL 356
24.5 IDENTICAL OR ALTERNATING IMAGES IN A CORNER 359
24.6 CYLINDER IN A UNIFORM ELECTRIC FIELD 361
24.7 RECIPROCAL POINT AND FIRST CIRCLE THEOREM (KIRCHHOFF, 1845) 363
24.8 INDUCED ELECTRIC CHARGES ON A CYLINDER 365
24.9 CHARGE NEAR INTERFACE BETWEEN TWO DIELECTRICS 368
25 SINGULAR SERIES OF ASCENDING-DESCENDING POWERS 375
25.1 LEMMA OF THE EXTREMA AND DOUBLY-CONNECTED CHAIN 375
25.2 ASCENDING AND DESCENDING GEOMETRIC SERIES 378
25.3 TOTAL CONVERGENCE IN A CLOSED SUBREGION 379
25.4 ABSOLUTE CONVERGENCE IN AN OPEN REGION 380
25.5 SERIES OF TEIXEIRA (1900): COEFFICIENTS AND REMAINDER 380
25.6 RESTRICTION TO LAURENT (1843)-WEIERSTRESS (1841) AND
LAURENT-MACLAURIN SERIES 381
25.7 HIERARCHY OF POWER SERIES EXPANSIONS 382
25.8 COEFFICIENTS OF REVERSION OF SERIES TO THIRD-ORDER 383
25.9 BINOMIAL EXPANSION AND SERIES AND INVERSE POWERS 384
26 MAGNETOSTATICS, CURRENTS, AND PERMEABILITY 389
26.1 MAGNETIC FIELD, INDUCTION, AND POLARIZATION (MAXWELL, 1873) 389
26.2 MAGNETIC PERMEABILITY, SUSCEPTIBILITY, AND FIELD FUNCTION 390
26.3 ELECTRIC CURRENT AND MAGNETIC FORCE (BIOT-SAVART) 391
26.4 HYDRODYNAMIC, ELECTROMAGNETIC, AND GRAVITY MULTIPOLES 392
26.5 CURRENT NEAR CONDUCTING OR INSULATING PLANE 394
26.6 IMAGE ELECTRIC CURRENTS IN A CORNER 396
26.7 CYLINDER IN A MAGNETIC FIELD OR NEAR A LINE-CURRENT 397
26.8 CURRENT NEAR CYLINDRICAL MAGNETIC INTERFACE 400
26.9 INFINITE MAGNETIC DIPOLE DISTRIBUTION 407
IMAGE 8
CONTENTS XIII
27 CLASSIFICATION OF SINGULARITIES AND FUNCTIONS 413
27.1 CHAIN OF INCLUSION OF REAL FUNCTIONS 413
27.2 SET OF COINCIDENCES FOR COMPLEX FUNCTIONS 416
27.3 ORDINARY POINTS, ZEROS, AND SINGULARITIES 418
27.4 RESIDUES AT POLES AND ESSENTIAL SINGULARITIES 420
27.5 INVERSION OF THE ORIGIN AND SINGULARITY AT INFINITY 421
27.6 IDENTIFICATION OF CONSTANTS (CAUCHY, 1844; LIOUVILLE, 1847) 423
27.7 DEFINITION OF POLYNOMIAL AND RATIONAL FUNCTION 424
27.8 ESSENTIAL SINGULARITY AS AN ACCUMULATION OF POLES 426
27.9 INTEGRAL, MEROMORPHIC, AND POLYMORPHIC FUNCTIONS 428
28 FORCES AND MOMENTS ON BODIES 439
28.1 KINETIC, ELECTRIC, MAGNETIC, AND GRAVITY ENERGIES 439
28.2 DRAG/THRUST, LIFT/DOWNFORCE, AND PITCHING MOMENT (KUTTA, 1902A;
JOUKOWSKI, 1906; BLASIUS, 1910) 442
28.3 HYDRODYNAMIC, ELECTROMAGNETIC, AND GRAVITY FORCES 453
28.4 FAIRING DUE TO A SOURCE OR SINK IN A STREAM (RANKINE, 1871) 460
28.5 OVAL/VALLEY/THROAT DUE TO A SOURCE AND SINK PAIR 462
28.6 VIRTUAL MASS OF A CYLINDER AND CAVITATION 466
28.7 FLOW PAST A CYLINDER WITH CIRCULATION 472
28.8 MOVING VORTEX AND SOURCE/SINK IMAGE SYSTEM 479
28.9 DIPOLE OUTSIDE OR INSIDE A CYLINDER 485
29 COMBINED TEST OF CONVERGENCE 493
29.1 BEHAVIOR OF SERIES AT ALL POINTS OF THE COMPLEX PLANE 493
29.2 CAUCHY (1821) NECESSARY AND SUFFICIENT CONDITIONS 498
29.3 REGION OF CONVERGENCE AND D ALEMBERT S RATIO (1768) 501
29.4 CONVERGENCE OF INTEGRALS AND HARMONIC SERIES 502
29.5 GAUSS TEST (1812A) AND EULER (1735)~MASCHERONI (1790) CONSTANT 504
29.6 CRITERIA AND SUMS OF ABEL (1826, 1839)-DIRICHLET (1862) 506
29.7 BOUNDARY OF CONVERGENCE AND WEIERSTRASS K-TEST (1856) 508
29.8 RADIUS AND EXPONENT OF A POWER SERIES 510
29.9 GAUSSIAN OR THREE-PARAMETER HYPERGEOMETRIC SERIES (GAUSS, 1812B)
511
30 EXAMPLES 30.1 TO 30.20 515
PART 4 CONFORMAL MAPPING: HEAT AND AERODYNAMICS 541
31 ANALYTIC CONTINUATION AND RATIONAL FUNCTIONS 543
31.1 THEOREM OF MONODROMY AND LACUNARY FUNCTIONS (OSGOOD, 1929) 543
31.2 CONJUGATE PROPERTY AND REFLECTION PRINCIPLE (RIEMANN, 1863;
SCHWARTZ, 1890) 547
31.3 ANALYTIC EXTENSION WITH JUMP ACROSS AN ARC (PLEMELJ, 1908) 550
31.4 THE CAUCHY (1821) FOURTH INTEGRAL THEOREM 553
31.5 NUMBER OF ZEROS AND POLES OF A FUNCTION 554
31.6 THEOREM OF ROUCHE (1858) AND FUNDAMENTAL THEOREM OF ALGEBRA 555
IMAGE 9
XIV CONTENTS
31.7 LEGENDRE S THEOREM AND ROOTS OF POLYNOMIALS 558
31.8 RATIONAL FUNCTIONS AND SIMPLE FRACTIONS 560
31.9 DECOMPOSITION INTO PARTIAL FRACTIONS AND RATIONAL INTEGRALS 563
32 STEADY HEAT CONDUCTION 567
32.1 HEAT FLUX AND THERMAL CONDUCTIVITY (FOURIER, 1818) 567
32.2 REGULARITY, ASYMPTOTIC, AND BOUNDARY CONDITIONS 570
32.3 IRROTATIONAL AND SOLENOIDAL POTENTIAL FIELDS 571
32.4 CORNER WITH ISOTHERMAL OR ADIABATIC WALLS 572
32.5 SOLID CYLINDER AND CYLINDRICAL CAVITY 574
32.6 HOLLOW TUBE WITH THICK OR THIN WALLS 578
32.7 CONVECTIVE TRANSFER IN HEAT EXCHANGERS 582
32.8 CONCENTRIC CYLINDERS OF DIFFERENT MATERIALS 586
32.9 PARALLEL WALLS OF AN INHOMOGENEOUS SUBSTANCE 589
33 CONFORMAL AND CRITICAL POINTS 595
33.1 PRESERVATION OF MODULUS AND DIRECTION OF ANGLES 595
33.2 INVERSION OF ANGLES AND ISOGONAL MAPPING 596
33.3 TRANSFORMATION OF ANGLES, LENGTHS, AND AREAS 597
33.4 CRITICAL POINTS OF THE FIRST AND SECOND KINDS 600
33.5 MULTIPLICATION AND DIVISION OF ANGLES INTO EDGES 602
33.6 INTERIOR POLYGONAL TRANSFORMATION (CHRISTOFFEL, 1868; SCHWARTZ,
1868) 604
33.7 INTERIOR AND EXTERIOR MAPPINGS AND POINT-AT-INFINITY 607
33.8 MAPPING OF A DISK INTO THE INTERIOR OF A POLYGON 609
33.9 FINITE INTERIOR AND OVERLAPPING EXTERIOR 610
34 WING SECTIONS AND PLANFORMS 615
34.1 FLOW PAST A FLAT PLATE AND KUTTA (1902B) CONDITION 616
34.2 JOUKOWSKI (1910) TRANSFORMATION AND THE ELLIPTIC CYLINDER 623
34.3 CIRCULAR ARC AND SYMMETRIC AIRFOILS 628
34.4 CAMBERED OR UNSYMMETRIC JOUKOWSKI (1916) AIRFOIL 632
34.5 PARAMETRIC FAMILIES AND GENERIC AIRFOILS (VON KARMAN TREFFTZ, 1918;
VON MISES, 1920; CARAFOLI) 637
34.6 LIFT AND PITCHING MOMENT AXIS AND COEFFICIENTS 643
34.7 SPANWISE DISTRIBUTION OF CIRCULATION ALONG A LIFTING-LINE (PRANDTL,
1918) 651
34.8 UNIFORM DOWNWASH AND ELLIPTIC LOADING 660
34.9 INDUCED, FORM, AND TOTAL DRAG 663
5 LINEAR AND HOMOGRAPHIC TRANSFORMATIONS 669
35.1 ROTATION, TRANSLATION, AND ISOMETRIC MAPPINGS 669
35.2 GROUP OF LINEAR MAPPINGS AND HOMOTHETY 671
35.3 ATTRACTIVE, REPULSIVE, AND INDIFFERENT LIMIT POINTS 672
35.4 UNIVALENT MAPPING AND HOMOGRAPHIC TRANSFORMATION 673
35.5 BILINEAR GROUP (MOBIUS) AND SELF-INVERSE FUNCTION 675
35.6 FOUR-POINT CROSS-RATIO AND FIXED POINTS 677
35.7 RECIPROCAL POINTS WITH REGARD TO THE CIRCLE AND THE STRAIGHT LINE
678
35.8 MAPPING OF A HALF-PLANE INTO A UNIT DISK 683
35.9 MAPPING BETWEEN INTERIORS AND EXTERIORS OF CIRCLES 684
IMAGE 10
CONTENTS XV
36 CHANNELS, CONDENSERS, AND WAKES 689
36.1 ROUNDED WEDGE AND CYLINDRICAL INDENTATION 690
36.2 IDENTICAL/ALTERNATING IMAGES FOR IRROTATIONAL/SOLENOIDAL FIELDS 697
36.3 PATH OF A MONOPOLE PAST A SHARP EDGE 704
36.4 CIRCULATION AROUND A FLAT PLATE AND FLOW THROUGH A SLIT 713
36.5 CONVERGENT CHANNEL (HARRIS, 1901) AND ADDED LENGTH 720
36.6 MONOPOLE IMAGES ON PARALLEL WALLS 727
36.7 CONFINED VORTEX AND SINGLE VORTEX ROW 731
36.8 SOURCE/SINK IN A WELL, ON A WALL OR AT A CORNER 737
36.9 PARALLEL AND STAGGERED DOUBLE VORTEX STREET (VON KARMAN, 1911;
LAMB, 1932) 743
37 MAPPING OF DOMAINS AND BOUNDARIES 755
37.1 UNICITY OF MAPPINGS AND BOUNDS IN THE UNIT DISK (SCHWARTZ, 1890;
CARATHEODORY, 1912; BOREL, 1928) 756
37.2 EXISTENCE OF A UNIFORMLY CONVERGENT SUBSEQUENCE
(VITALI, 1903; MONTEL, 1910; OSGOOD, 1929) 759
37.3 SIMPLY CONNECTED REGION WITH AT LEAST TWO BOUNDARY POINTS (RIEMANN,
1863) 762
37.4 MAPPING BETWEEN MULTIPLY CONNECTED REGIONS BY
MULTIVALENT/MULTIVALUED FUNCTIONS 765
37.5 MINIMAX, REFERENCE, AND PUNCTURED MAPPINGS 770
37.6 AUTOMORPHISM GROUP AND FUNDAMENTAL REGIONS 772
37.7 CORRESPONDENCE OF INTERIORS AND BOUNDARIES FOR COMPACT AND
NONCORNPACT REGIONS 775
37.8 INTERIOR AND EXTERIOR INTEGRAL THEOREMS (CAUCHY, 1821; SCHWARTZ,
1890) 780
37.9 HARMONIC FUNCTIONS DEFINED BY BOUNDARY VALUES
(POISSON, 1820; DIRICHLET, 1850; ROBIN, 1886;
VON NEUMANN, 1961) 783
38 HODOGRAPH FOR FREE JETS 797
38.1 FIELDS DUE TO POTENTIALS ON PLANES AND CYLINDERS 797
38.2 WIDTH OF THE VENA CONTRACTA OF A JET (BORDA, 1766) 805
38.3 SLIT IN A WALL AND REENTRANT TUBE IN A RESERVOIR
(HELMHOLTZ, 1868) 807
38.4 FLAT PLATE ORTHOGONAL TO A JET OR TO A WALL (KIRCHHOFF, 1869;
RAYLEIGH, 1876A) 812
38.5 CENTER OF PRESSURE AND DIVIDING STREAMLINE ON A SURFBOARD
(RAYLEIGH, 1876B, 1891) 817
38.6 ARROW OR BENT LAMINA IN A STREAM (RETHY, 1879;
BOBYLEFF, 1881) 824
38.7 JET ATTACHMENT AROUND A WALL (COANDA EFFECT) 830
38.8 FLUIDICS: DEFLECTION OF A JET BY A SMALL SOURCE 834
38.9 JETS MERGING, SPLITTING, OR COLLIDING WITH A WALL 839
39 ESSENTIAL SINGULARITIES, ROOTS, AND PERIODS 853
39.1 CLASSIFICATION OF SPECIAL AND SINGULAR POINTS 853
39.2 ZEROS, POLES, AND ESSENTIAL SINGULARITIES (CASORATI, 1868;
WEIERSTRASS, 1876; PICARD, 1880) 856
39.3 EXCEPTIONAL VALUE (PICARD, 1879) AND INFINITE NUMBER OF ROOTS 860
IMAGE 11
XVI CONTENTS
39.4 TRIANGULAR COVERINGS (SCHWARTZ, 1890) AND THE MODULAR FUNCTION
(LEGENDRE) 864
39.5 DENSE RAYS AND JULIA (1924) THEOREM 870
39.6 CAUCHY BOUNDS AND LANDAU (1904) RADIUS (SCHOTTKY, 1904;
CARATHEODORY, 1912; MONTEL, 1927) 875
39.7 PERIODS AND INVERSION OF HYPERELLIPTIC INTEGRALS 882
39.8 LOGARITHM, EXPONENTIAL, AND CIRCULAR/HYPERBOLIC FUNCTIONS 886 39.9
ELLIPTIC FUNCTIONS OF JACOBI (1827) AND WEIERSTRASS (1895) 889
40 EXAMPLES 40.1 TO 40.20 897
BIBLIOGRAPHY 953
REFERENCES 963
INDEX 969
|
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| author | Campos, Luis Manuel Braga da Costa 1950- |
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| language | English |
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| spelling | Campos, Luis Manuel Braga da Costa 1950- Verfasser (DE-588)142675075 aut Complex analysis with applications to flows and fields L. M. B. C. Campos Boca Raton, Fla. [u.a.] CRC Press 2011 XL, 989 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and physics for science and technology Functions of complex variables Anwendung (DE-588)4196864-5 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s Anwendung (DE-588)4196864-5 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022469444&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Campos, Luis Manuel Braga da Costa 1950- Complex analysis with applications to flows and fields Functions of complex variables Anwendung (DE-588)4196864-5 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4196864-5 (DE-588)4018935-1 (DE-588)4123623-3 |
| title | Complex analysis with applications to flows and fields |
| title_auth | Complex analysis with applications to flows and fields |
| title_exact_search | Complex analysis with applications to flows and fields |
| title_full | Complex analysis with applications to flows and fields L. M. B. C. Campos |
| title_fullStr | Complex analysis with applications to flows and fields L. M. B. C. Campos |
| title_full_unstemmed | Complex analysis with applications to flows and fields L. M. B. C. Campos |
| title_short | Complex analysis with applications to flows and fields |
| title_sort | complex analysis with applications to flows and fields |
| topic | Functions of complex variables Anwendung (DE-588)4196864-5 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Functions of complex variables Anwendung Funktionentheorie Lehrbuch |
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