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| Ïîèñê ïî óêàçàòåëÿì |
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| Kaplan W. — Introduction to analytic functions |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel summation of series 144—145 (Prob. 5)
Abel’s theorem 65
Absolute convergence 10 172 176 178 180 182 184
Absolute value 2
Addition 2
Airy’s stress function 148
Algebraic branch point 162
Algebraic equation 115 (Prob. 5)
Amplitude 2
Analytic at a point 39
Analytic at infinity 94
Analytic continuation 49 75 159—163 166 189—195 201—202
Analytic function 35 54 74 123 166—167
Analytic, along a curve 39
angle 2 4—5
Annular domain 89 184—185
Arc length 31
Argument of complex number 2 54—55
Argument, Principle 112 197
Arithmetic mean 72 82
Associated radii of convergence 184 (Probs. 7 8)
Average value 72 82
Biharmonic equation and functions 148—150 151
Boundary, conditions 136—137
Boundary, points 14 91 168
Boundary, value problem 135—145
Bounded function 77 (Probs. 7 8) 201
Branch 53—61 159
Branch, point 54 59 60 162
Cauchy — Riemann equations 27—28 (Probs. 9 10) 33 36 83 146 175
Cauchy, criterion 9
Cauchy, inequalities 77 (Prob. 7)
Cauchy, integral formula 71—72 170
Cauchy, integral theorem 42—43 69—70 74
Cauchy, Residue Theorem 102
Change of variable in integrals 45—46
Circle of convergence 64 75
Closed poly cylinder 169
Comparison test for series 11
Complete Reinhardt domain 179 183 192 195
Complex conjugate 2
Complex n-dimensional space 166
Complex number infinity 9—10 18 94—96 107
Complex number system 1—4
Conformal mapping 122—158
Conjugate harmonic functions 79
Continuity 16—18 167
convergence of integrals 119
Convergence of sequences 8—9 19
Convergence of series 10—12 19
Convergence region of power series 62 176—184
Convex 125 181 192
Convex, hull 191 194—195 4)
Critical points 133 (Prob.7)
Cross ratio 133—134 (Prob.9)
Curl of a vector field 83 146
De l’Hopital’s rule 115
De Moivre theorem 5
Definite integral 116—121
Degree of rational function 98—99 (Prob.9)
Deleted neighborhood 90
Derivative 22—28
Differential 24
Differential equation 68 (Prob.4)
Differentiation of integrals 44—45 47 (Prob.4) 85 171
Differentiation of series 66—67
Directional derivative 23 39—40
Dirichlet problem 135—145
Distance 3 166
Divergence of a vector field 83 146
Divergence of sequences, series, integrals (see Convergence)
Diverges to infinity 9
Domain 14 167
Domain of convergence 62 176
Double series 172 185
Doubly connected 69
Electrostatic potential 143 (Prob.2)
Elementary analytic functions 47—51
Entire function 77 (Prob.8)
Envelope of holomorphy 191—195
equation (see Algebraic equation)
Equilibrium 143 (Prob.3)
Equipotential lines 146
Error, estimation of 31 75
Essential singularity 92 95 201
Euler identity 6
Exponential function 6 48 50 58 129—130
Exponential function, transformation 129—130
Extended z plane 96
Finite z plane 96
Flow past an obstacle 147
Fluid Motion (see Hydrodynamics)
Fourier, integral 119
Fourier, series 51 64—65 83 144 5)
Function 14—16 53—54 122
Function of several complex variables 165
Functional equation, permanence of 49 161
Fundamental theorem of algebra 115 (Prob.5)
Generalized polycylindrical domain 168
Geometric series 19—21 64
Harmonic, conjugate 79
Harmonic, function iii-iv 22—23 33 35 79—83 135—145
Holomorphic function 167
Hydrodynamics 146—148 151 2)
Hyperbolic functions 48—51
Imaginary part 2
Incompressible flow 146—148 151 2)
Indefinite integral 44 48
Independence of path 43
Indirect analytic continuation 160 190
Inequality 3 31 77
Infinite, sequences 8—12 19
Infinite, series (see also Fourier series Power
Infinity, complex 9—10 18 94—96 107
Integral, complex 28—32
| Integral, depending on a parameter 85 (Prob. 9)
Integral, function 77 (Prob. 8)
Integration of series 32
Interior points of a sot 178—179
Inverse, functions 53—54 84 123 124
Inverse, logarithmic image 191 195
Inverse, points 134 (Prob.10)
Inverse, trigonometric functions 60
Irrotational 146
Isolated, boundary point 91
Isolated, singularity 91 194
Isolation of zeros 93 190 195
Iterated series 173
Jacobian determinant 112
Kronecker integral 111
Laplace equation (see Harmonic functions)
Laurent series 88—121 184—189
Leibnitz rule for integrals 47 (Prob. 4) 85 171
Level curves 124 128—131
LIMIT 8—10 16—19 167
Line integral 29
Linear, fractional transformation 126
Linear, integral transformation 126
Linear, transformation 199
Liouville’s theorem 77 (Prob.8) 203
Logarithmic, branch point 54
Logarithmic, convexity 181 183 192
Logarithmic, derivative 110
Logarithmic, function 55—57
Logarithmic, image 181 184
Logarithmic, potential 86 (Prob. 9)
Logarithmic, residue 110
M-test for series 20
M-uniformly convergent 173 179 185
Mapping (see Conformal mapping Transformation)
Modulus 2
Morera’s theorem 44
Multiplication 2—7
Multiplication of HorioH 12
Multiplicity of zero 93 95
Multiply connected 69 72
Neighborhood 14 90 167
Nonessential singularity 201 203
Nonessential singularity wth-term test 11
One-to-one transformation 112—113 122—125 152
Open region 14 167
Order of pole 92 95
Order of zero 93 95
Partial, derivative 167 171
Partial, sum of series 10
Piecewise smooth path 28
Poisson, equation 150
Poisson, integral formula 80—86 130—137 140 144—145
Poisson, summation of series 144—145 (Prob.5)
Polar form of complex numbers 4
Pole 92 95 103 201
Polycylindrical domain 108
Polynomial 17—18 48 167
Power, function 5—7 58—59
Power, series 62—87 88—90 172 176 180
Prime 200 202
Principal, part 92
Principal, value of function 56 58
Principal, value of integral 119
Prob. 11
Pseudoalgebraic equation 196
Pseudopolynomial 196—200
Pure imaginary 2
Radius of convergence 62—63 75
Ratio Test 11 63—64
Rational function 18 48 98 114 4) 126—127
Real part 2
Rearrangement of series 172—173 176
Reciprocal transformation 126
Reinhardt domain 178 181 184 185 192 195 6)
Removable singularity 91 94 99 194 195 200—203
Residues 101—121
Riemann, surfaces 162—163
Riemann, theorem of 99 (Prob.12) 201
Root test 11
Roots 5—7 16 58—60 115
Rotation-stretching 125—126
Schwarz-Christoffel transformation 154
Sense-preserving 122
Sequences (see Infinite sequences)
Series (see Infinite series)
Simply connected 42—43 71 80
Singular point 91 200—203
Slit domain 147
Smooth path 28
Stationary flow 146—148
Stereographic projection 95
Stream function and lines 146
Stress function and tensor 148
Subtraction 2—3
Sum of numbers 2
Sum of series 10
Tangent vector 26 (Prob.7) 123 135
Taylor, formula with remainder 75
Taylor, series 35 67 73—74 172—180
Temperature 143 (Prob. 3) 151
Tensor 148
Transformation 15 112—113 122—158 199
Translation 125
Triangle inequality 3
Trigonometric, functions 48—52
Trigonometric, series 51 64—65 83 144 5)
Triply connected 69
Uniform convergence 19—21 32 62 173
Uniqueness of analytic continuation 159 190
Vector field 15 83 146
Velocity potential 146
Weierstrass and Casorati, theorem of 99 (Prob. 13)
Weierstrass, M-test 20 173
Weierstrass, preparation theorem 196—200
Zeros of functions 16 52 93 95 98—99 9 10) 166 195—200
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