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| Bitsadze A.V. — Equations of Mathematical Physics |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel, N. H. 45 227
Abel’s integral equation 45 227
Adjoint equation 177 193
Adjoint operator 86 193
Admitted functions 303
Analytic continuation 146
Analytic function 100
Analytic function of one complex variable 98ff
Analytic function of several complex variables 165
Analytic function, of real variables 170
Angle-preserving property 106 173
Antiderivative 124
Area mass density 41
Asymptotic expansion 291
Bessel, W. F. 256 261 270 271
Bessel’s equation 259
Bessel’s equation of the first kind 259
Bessel’s equation of the second kind (or Neumann’s functions) 271
Bessel’s equation of the third kind (or Hankel’s functions) 270
Bessel’s equation, (cylindrical) functions 261
Bessel’s equation, inequality 256
Boundary condition 44
Boundary-value problem first (or Dirichlet’s) 34
Boundary-value problem first (or Dirichlet’s) with oblique derivative boundary condition 91
Boundary-value problem first (or Dirichlet’s), Poincar ’s 91
Boundary-value problem first (or Dirichlet’s), second (or Neumann’s) 83 91
Boundary-value problem first (or Dirichlet’s), third (mixed) 91
Branch of a many-valued function 113
Branch point 114
Bubnov — Galerkin method 310
Bubnov, I. G. 310
Cauchy — Dirichlet problem 204ff
Cauchy — Hadamard formula 103
Cauchy — Riemann partial differential equations for two independent variables 40 97
Cauchy — Riemann partial differential equations for two independent variables for several independent variables 165
Cauchy, A. L. 40 97 103 118 150 165 168
Cauchy-type integral 124
Cauchy’s for Laplace’s equation 192
Cauchy’s, general 188ff
Cauchy’s, integral formula 121 168
Cauchy’s, principal value of a singular integral 150ff
Cauchy’s, problem (or initial-value problem) 34
Cauchy’s, theorem 118ff
Characteristic curve 21
Characteristic determinant 19
Characteristic direction 21
Characteristic form 15
Characteristic value (number) 217
Chebyshev, P. L. 261
Chebyshev’s equation 261
Chebyshev’s equation, functions 261
Chebyshev’s equation, polynomials 262
Circle of convergence 103
Circulation 38
Closed contour negatively oriented 116
Closed contour negatively oriented, positively oriented 116
Coefficient of thermal conductivity 43
Coiiformal mapping 108
Coiiformal mapping of a circle onto a circle 111
Coiiformal mapping of a half-plane onto a circle 111
Coiiformal mapping of a half-plane onto a half¬plane 110
Coiiformal mapping, angle-preserving property of 106 173
Coiiformal mapping, Gauss 173
Coiiformal mapping, in Euclidean space 172ff
Complex integration 116ff
Complex integration, of a power function 119ff
Complex vector space 163
Composite function 108
Composition formula for singular integrals 240
Condition for solvability of Neumann problem 83
conformal 87
Conjugate harmonic functions 126
Continuity principle for analytic functions 147
Convolution (or German faltung) 281
Derivative(s) of analytic function of one variable 101
Derivative(s) of analytic function of one variable of a volume potential 67ff
Derivative(s) of analytic function of one variable of analytic function of several variables 166
Derivative(s) of analytic function of one variable, normal, of a double layer potential 80
Derivative(s) of analytic function of one variable, normal, of a single-layer potential 82
Derivative(s) of analytic function of one variable, tangential, of a single-layer potential 151
Derivative(s) of analytic function of one variable, total, of an analytic function of several variables 165
Differential of an analytic function of one variable 101
Dipole 75
Dipole, axis of 75
Dipole, polarization (or dipole moment) of 75
Dirac, P. A. M. 284
Dirac’s delta function 284ff
Dirac’s delta function, principle 305
Dirac’s delta function, problem (or first boundary-value problem) 34 91
Dirichlet, P. G. L. 34 42 91 303
Dirichlet’s integral 142 303
Domain of dependence 181
Domain of dependence of influence 182
Domain of dependence of propagation of solution of wave equation 182
D’Alembert, J. 180
D’Alembert’s formula 180
Eigenfunction 218 248
Eigenvalue 248
Eigenvalue problem 305ff
Eigenvalue problem, transcendental 141
Elliptic partial differential equations 16 18 47ff
Entire function 141
Equation of oscillation of a membrane 251
Equation of oscillation of a membrane of a string 180 246
Equation of oscillation of a membrane, ff 264
Equations of mathematical physics 37
Essential singular point of an analytic function 138
Euler, L. 41 50 303
Euler’s equation 41 303
Euler’s equation, gamma function 50 272
Exponential function 114
Extended complex plane 98
Extremum principle for elliptic partial differential equations 92
Extremum principle for harmonic functions 53
Extremum principle for heat conduction equation 200
Field intensity 38
Finite-difference approximation of a differential equation 286ff
Flux through a contour 38
Force function of a field 39
Fourier, J. B. J. 43 273
Fourier’s coefficients 255
Fourier’s coefficients, inverse 273
Fourier’s coefficients, inverse transformation 273
Fourier’s coefficients, law of heat conduction 43
Fourier’s coefficients, method (or method of separation of variables) 246ff
Fourier’s coefficients, transform of a function 273
Fourier’s coefficients, transformation 273
Fredholm, E. I. 36 210 217 218 219 221
Fredholm’s of the first kind 36
Fredholm’s of the second kind 36 210
Fredholm’s of the third kind 36
Fredholm’s, alternative 221
Fredholm’s, first theorem 217
Fredholm’s, integral equation 36
Fredholm’s, second theorem 218
Fredholm’s, third theorem 219
Function(s) analytic 100
Function(s) of a complex variable 98
Function(s) of a complex variable, continuous 99
Function(s) of a complex variable, monogenic 99
Function(s), conjugate 126
Function(s), exponential 114
Function(s), Hankel’s (or Bessel’s, of the third kind) 270
Function(s), harmonic 26 47
Function(s), infinite-sheeted 115
Function(s), infinite-valued 115
Function(s), inverse 105 113
Function(s), limit of 99
Function(s), logarithmic 113
Function(s), many-sheeted 113
| Function(s), many-valued 113
Function(s), meromorphic 141
Function(s), metaharmonic 259
Function(s), monogenic 99
Function(s), Neumann’s (or Bessel’s, of the second kind) 271
Function(s), one-valued 98
Function(s), power 111
Function(s), special 261
Function(s), univalent (or one-sheeted) 104
Fundamental (or elementary) solution 14
Fundamental (or elementary) solution of heat conduction equation 33
Fundamental (or elementary) solution) of Laplace’s equation 27
Fundamental (or elementary) solution), principal 94
Galerkin, B. G. 310
Gauss — Ostrogradsky formula 31
Gauss, K. F. 31 73 173
Gauss’ formula 73
Generalized potential function for a double layer 96
Generalized potential function for surface distribution of mass 95
Generalized potential function for volume distribution of mass 89
Goursat problem 192
Goursat, E. 192
Green, G. 55 87 234
Green’s function 55ff 234
Green’s function, symmetry of 55
Green’s Theorem 87
Grid-point 286
Grid-size 286
H lder, O. 150
Hadamard, J. S. 103 192
Hamilton, R. W. 26 41
Hamiltonian operator (or nabla or del) 26
Hamilton’s principle 41
Hankel, H. 270
Hankel’s function (or Bessel’s function of the third kind) 270
Harmonic function(s) 26 47
Harmonic function(s), conjugate 126
Harmonic function(s), regular at infinity 48
Harmonic polynomial 28
Harnack, A. 64
Harnack’s theorem 64
Hartogs, F. 166
Hartogs’ theorem 166
Heat conduction equation 32 200ff
Heat conduction equation, non-homogeneous 208
Helmholtz equation 94 252
Helmholtz, H. L. F. 94 252
Hilbert, D. 240
Hilbert’s integral equation 240
Hilbert’s integral equation, inverse transformation 241
Hilbert’s integral equation, on a finite interval 243
Hilbert’s integral equation, transformation 241
Holder’s condition for a function of one variable 150
Holder’s condition for a function of one variable for a function of several variables 236
Huygens, C. 178
Huygens’ principle 178
Hyperbolic partial differential equation 16 18 176ff
Image of a point (under a mapping) 105
Image, (or transform) of a function under an integral transformation 272
Improperly posed (or not well- posed) problem 33 192
Indefinite integral 123
Infinite-sheeted function 115
Infinite-valued function 115
Initial conditions 44
Integral equation 35ff 210ff
Integral equation with kernel having logarithmic singularity 244
Integral equation, Abel’s 45 227
Integral equation, Fredholm’s 36
Integral equation, Fredholm’s, of the first kind 36
Integral equation, Fredholm’s, of the second kind 36 210
Integral equation, Fredholm’s, of the third kind 36
Integral equation, Fredholm’s, with degenerate kernel 215ff
Integral equation, Hilbert’s 240
Integral equation, linear 35
Integral equation, linear, homogeneous 35
Integral equation, linear, non-homogeneous 35
Integral equation, singular 236
Integral equation, singular, of the theory of the wing of an airplane 242
Integral equation, singular, with logarithmic singularity 244
Integral equation, Volterra’s 37
Integral equation, Volterra’s, of the first kind 226
Integral equation, Volterra’s, of the second kind 37 211
Integral equation, Volterra’s, with multiple integral 225
Integral operator 35
Integral representation of a harmonic function 50
Inverse function 105 113
inversion 84 175
Isolated singular point 138
Isolated singular point, at infinity 140
Iterative method (or method of successive approximations) 210ff
Jordan curve 76
Jordan, C. 76
Kernel of an integral equation 35
Kernel of an integral equation, Cauchy’s 240
Kernel of an integral equation, degenerate 215
Kernel of an integral equation, Hilbert’s 240
Kernel of an integral equation, iterated 219
Kernel of an integral equation, resolvent 219
Kernel of an integral equation, symmetric 224
Kirchhoff’s formula 177
Kirehhoff, G. 177
Lagrange adjoint operator 267
Lagrange, J. L. 267
Laguerre, E. N. 262
Laguerre’s equation 262
Laplace, P. S. 26 263 272
Laplace’s equation 26
Laplace’s equation, inverse 272
Laplace’s equation, nverse transform (or inverse image or the original) 272
Laplace’s equation, operator 26
Laplace’s equation, surface (spherical) harmonics 263
Laplace’s equation, transform (or image) of a function 272
Laplace’s equation, transformation 272
Laurent, P. M. H. 135
Laurent’s expansion (series) 136
Laurent’s expansion (series), principal (or singular) part of 137
Laurent’s expansion (series), regular part of 137
Laurent’s expansion (series), theorem 135
Lebesgue, H. 49
Lebesgue’s integral 49
Legendre, A. M. 264
Legendre’s equation 264
Legendre’s equation, associated 264
Legendre’s equation, functions 264
Leibniz, G. W. 124
Limiting values of Cauchy-type integral 154ff
Linear partial differential equation 14
Linear-fractional function 102
Liouville, J. 63 134 247
Liouville’s theorem for analytic functions 134
Liouville’s theorem for analytic functions for harmonic functions 63
Lyapunov curve 158
Lyapunov, A. M. 158
Maximum Modulus Principle 127
Mean-square deviation 255
Mean-value formula for a ball 53
Mean-value formula for a ball for a sphere 53
Mellin, R. H. 274
Mellin’s, inverse 275
Mellin’s, transform of a function 275
Mellin’s, transformation 274
Meromorphic function 141
Metaharmonic equation 259
Metaharmonic function 259
Method(s) of separation of variables (or Fourier’s method) 246ff
Method(s), finite-difference 286ff
Method(s), of integral transformations 266ff
Method(s), saddle-point 301
Method(s), variational 303
Minimizing sequence 307
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