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| Wrede R.C., Spiegel M. — Theory and problems of advanced calculus |
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| Предметный указатель |
Dirichlet's test, for series 270 303
Discontinuities 47 119
Discontinuities, removable 56 119
Distance between points 165
Distributive law 2
Distributive law, for cross products 154
Distributive law, for dot products 153
Divergence 158 159 172—174
Divergence theorem 236 249—252 261
Divergence theorem, proof of 249 250
Divergence, in curvilinear coordinates 161
Divergence, in cylindrical coordinates 161
Divergence, of improper integrals 306—309 see
Divergence, of infinite series see "Infinite series"
Divergent integrals 306—335
Divergent integrals, sequences 23 see
Divergent integrals, series 25 see
Division 1
Division by zero 8
Division, of complex numbers 6 7
Domain, of a function 39 116
Domain, of convergence 272
dot products 153 154 165 166
Dot products, commutative law for 153
Dot products, distributive law for 153
Dot products, laws for 153 154
Double series 277
Dummy variable 94
Duplication formula for gamma function 286 378 386
e 4
Electric field vector 181
Electromagnetic theory 181
Elementary transcendental functions 43 71 95
Elements, of a set 1
ellipse 114
Ellipse, area of 114
Empty set 1
Envelopes 185 186 192
Equality, of complex number 6
Equality, of vectors 158
Equations, difference 65
Equations, integral 364 369 370
Equations, polynomial 6 43
Equipotential surfaces 186
Errors, applications to 189 200 204
Errors, in computing sums of alternating series 266 282
Errors, mean square 353
Essential singularity 395
Eudoxus 90
Euler's, constant 296 378 388
Euler's, formulas or indentities 8 295
Euler's, theorem on homogeneous functions 122
Euler, Leonhart 336
Even function 338 347—351
Everywhere dense set 2
Evolute 185
Exact differentials 122 131 132 231 see
Expansion of functions, in Fourier series see "Fourier series"
Expansion of functions, in power series 272
Expansions see "Series"
Explicit functions 123
Exponential function 42
Exponential order 334
Exponents 3 11
Factorial function see "Gamma functions"
Fibonacci sequence 35 37
Field 2
Field, conservative 233
Field, scalar 153
Field, vector 153
Fluid mechanics 402
Fourier coefficients 337 345
Fourier coefficients, expansion see "Fourier series"
Fourier integrals 363—374 see
Fourier integrals, convergence of see "Fourier's integral theorem"
Fourier integrals, solution of boundary-value problems by 371
Fourier series 336—362
Fourier series, complex notation for 339
Fourier series, convergence of 338 354—356
Fourier series, differentiation and integration of 339
Fourier series, Dirichlet conditions for convergence of 337
Fourier series, half range 338 347—351
Fourier series, Parseval's identity for 310 338 351 352
Fourier series, solution of boundary-value problems by 339 356—358
Fourier transforms 364—368 see
Fourier transforms, convolution theorem for 365
Fourier transforms, inverse 364
Fourier transforms, Parselval's identities for 366 368 373
Fourier transforms, symmetric form for 364
Fourier's integral theorem 363 364
Fourier's integral theorem, heuristic demonstration of 369
Fourier's integral theorem, proof of 369
Fourier, Joseph 336
Fractions 1
Frenet — Serret formulas 159
Fresnel integrals 387
Frullani's integral 333
Functional determinant 123 136 see
Functional notation 39 116
Functions 39—64 116 132 392
Functions of a complex variable 392—423
Functions of a Complex variable, analytic 393
Functions of a complex variable, Cauchy — Riemann equations 393 400 see
Functions of a complex variable, continuity of 393 399 400
Functions of a complex variable, definition of 392
Functions of a complex variable, derivatives of 393 400—403
Functions of a complex variable, elementary 393
Functions of a complex variable, imaginary part of 392 400
Functions of a Complex variable, integrals of 394 403—406
Functions of a complex variable, Jacobians and 422
Functions of a complex variable, Laplace transforms and 423
Functions of a complex variable, limits of 393 399 400
Functions of a complex variable, line integrals and 394
Functions of a complex variable, multiple-valued 392
Functions of a complex variable, poles of 395
Functions of a complex variable, real part of 392 400
Functions of a complex variable, residue theorem for see "Residue theorem"
Functions of a complex variable, series of 286 395 406—409
Functions of a complex variable, single-valued 392
Functions of a complex variable, singular points of 395
Functions, algebraic 43
Functions, bounded 40 41
Functions, branches of 41
Functions, continuity of see "Continuity"
Functions, decreasing 41 42
Functions, definition of 39 116
Functions, derivatives of see "Derivatives"
Functions, differential of see "Differentials"
Functions, domain of 39 116
Functions, elementary transcendental 43 44
Functions, even 338 347—351
Functions, explicit and implicit 123
Functions, harmonic 393
Functions, hyperbolic 44 45
Functions, hypergeometric 276 303
Functions, increasing 41 42
Functions, maxima and minima of see "Maxima and minima"
Functions, monotonic 41
Functions, normalized 342
Functions, odd 238 347—351
Functions, of a function see "Composite function"
Functions, of several variables 116 123 126
Functions, orthogonal 342 357 358
Functions, orthonormal 342
Functions, periodic 336
Functions, polynomial 43
Functions, sequences and series of 269 270 272 286 289
Functions, single-valued 39 392
Functions, staircase of step 51
Functions, transcendental 43 44
| functions, types of 43 44
Fundamental theorem, of algebra 43
Fundamental theorem, of calculus 94 104
g.l.b. see "Greatest lower bound"
Gamma functions 375—391
Gamma functions, analytic continuation of 376
Gamma functions, asymptotic formulas for 376 378
Gamma functions, duplication formula for 378 386
Gamma functions, infinite product for 377
Gamma functions, recurrence formula for 375 376
Gamma functions, Stirling's formulas and asymptotic series for 378 384
Gamma functions, table and graph of 375
Gauss'  function 377
Gauss' differential equation 276
Gauss' test 268 283
Geometric integral 308
Gibbs, Williard 150
Gradient 158 161 162 172
Gradient, in curvilinear coordinates 161
Gradient, in cylindrical coordinates 162
Graph, of a function of one variable 41
Graph, of a function of two variables 144 145
Grassman, Herman 150
Greater than 3
Greatest limit see "Limit superior"
Greatest lower bound 6
Greatest lower bound of a function 41
Greatest lower bound of a sequence 24 32 36
Green's theorem in space see "Divergence theorem"
Green's theorem in the plane 232 240—243 260
Grouping method, for exact differentials 132
Half range Fourier sine or cosine series 238 239 347—351
Hamilton, William Rowen 150 158
Harmonic functions 393
Harmonic series 266
Heat conduction equation 357
Heat conduction equation, solution of, by Fourier integrals 369
Heat conduction equation, solution of, by Fourier series 354 355
Homogeneous functions, Euler's theorem on 122
Hyperbolic coordinates 218
Hyperbolic functions 44 45
Hyperbolic functions, inverse 41
Hyperboloid of one sheet 127
Hypergeometric function or series 276
Hypersphere 116
Hypersurface 116
Identity, with respect to addition and multiplication 2
Image or mapping 124 416
Imaginary part, of a complex number 6
Imaginary part, of functions of a complex variable 392 399 400
Imaginary unit 6
Implicit functions 69 123
Implicit functions and Jacobians 135—139
Improper integrals 97 110 114 306—335
Improper integrals, absolute and conditional convergence of 309 312 319—321
Improper integrals, comparison test for 308 311
Improper integrals, containing a parameter 313
Improper integrals, definition of 306
Improper integrals, of the first kind 306—308 317—321
Improper integrals, of the second kind 306 310—312 321 322
Improper integrals, of the third kind 306 313 322
Improper integrals, quotient test for 304 311 315
Improper integrals, uniform convergence of 313 314 323 324
Improper integrals, Weierstrass M test for 313 324—329
Increasing functions 41
Increasing functions, monotonic 41
Increasing functions, strictly 41 47
Increasing sequences, monotonic and strictly 24
Indefinite integrals 94 see
Independence of the path 212 213 243—245 260
Independent variable 39 116
Indeterminate forms 56 82—84
Indeterminate forms, L'Hospital's rules for see "L'Hospital's rules"
Induction, mathematical 8
Inequalities 3 10
Inequality 3
Inequality, Bernoulli's 16
Inequality, Cauchy's 422
Inequality, Schwarz's 10 18 110
Inferior limit see "Limit inferior"
Infinite product 277
Infinite product, for gamma function 376
Infinite series 25 33 37 265—305 see
Infinite series, absolute convergence of 268 283 300
Infinite series, comparison test for 267 279 280
Infinite series, conditional convergence of 268 283
Infinite series, convergence tests for 266—268
Infinite series, functions defined by 276
Infinite series, Gauss' test for 268
Infinite series, integral test for 267 280—283
Infinite series, nth root test for 268
Infinite series, of complex terms 276
Infinite series, of functions 269 270 276 277
Infinite series, partial sums of 25 266
Infinite series, quotient test for 267 278
Infinite series, Raabe's test for 268 285
Infinite series, ratio test for 268 284 300
Infinite series, rearrangement of terms in 269
Infinite series, special 270
Infinite series, uniform convergence of 269 270 see
Infinite series, Weierstrass M test for 270 289
Infinite, countably 5
Infinite, interval 5
Infinitesimal 89
infinity 25 46
Inflection, point of 74
Initial point, of a vector 150
Integers, positive and negative 1
Integrable 91
Integral equations 367 372 373
Integral formulas of Cauchy 394 403—406
Integral test for infinite series 267
Integrals 90—115 207—228 306—335 363—374 394 398 409—423 see
Integrals, definite 90 91 see
Integrals, Dirichlet 379 385 389
Integrals, double 207 213—219
Integrals, evaluation of 314 325—327 398 412—416
Integrals, Fresnel 387
Integrals, Frullani's 333
Integrals, indefinite 94
Integrals, iterated 208—210
Integrals, mean value theorems for 72 92
Integrals, of elementary functions 96
Integrals, of functions of a complex variable 392—423
Integrals, of infinite series of functions 272 275
Integrals, Schwarz's inequality for 110
Integrals, table of 96
Integrals, transformations of 95 103—108 299
Integrals, uniform convergence of 313 314 323 324
integrand 91
Integrating factor 223
Integration, applications of 98 109 110 114 see
Integration, by parts 97—102
Integration, contour 398
Integration, interchange of order of 209
Integration, limits, of 91
Integration, of elementary functions 96 97 107
Integration, of Fourier series 339 353
Integration, range of 91
Integration, special methods of 97 105—108
Integration, under integral sign 186 195
Intercepts 126
Interior point 117
Intermediate Value Theorem 48
Intersection of sets 12
Intervals, closed 5
Intervals, continuity in 47
Intervals, infinite 5
Intervals, nested 25 32
Intervals, of convergence 25
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