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Efimov A.V. — Mathematical Analysis (Advanced Topics). Part 1. General Functional Series and Their Applications
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Íàçâàíèå: Mathematical Analysis (Advanced Topics). Part 1. General Functional Series and Their ApplicationsÀâòîð: Efimov A.V. Àííîòàöèÿ: In the recent years mathematics has become a commonplace tool not only in disciplines such as mechanics, physics, chemistry, and astronomy, but also in ones such as economics, biology, medicine, and linguistics. This invasion of mathematics into every field of scientific and practical activity proceeds ever more intensely. We all witness the increasing of mathematization of sciences, a process aided by the rapid development of electronic computers. In order for most technical, economic, or biological problems to be solved, it is first necessary that they be "translated" into mathematical language, whence they can be solved. It is obvious that the most difficult link in this chain is the "translation" of a problem into mathematical language. This is because the correct mathematical statement of an engineering or other problem requires that the researcher possess both a knowledge of the science which gave birth to the problem, and a deep understanding of mathematics...
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñåðèÿ: Ñäåëàíî â õîëëå Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1985Êîëè÷åñòâî ñòðàíèö: 360Äîáàâëåíà â êàòàëîã: 12.10.2013Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Acceleration oj series convergence 146—152 Algorithm of fast Fourier’s transform 257—263 Analytic continuation 137 Argument 22 Argument of a complex number 11 Argument of a derivative 36 Argument of a derivative, geometrical meaning of 36 Argument of the product of two complex numbers 13 Argument of the quotient of two complex numbers 13 Basic frequency 203 Boundary correspondence principle 41 Cauchy — Riemann conditions 30 Cauchy’s criterion 81 103 Cauchy’s integral and its application 69—78 Circle of convergence 120 Coefficient(s), Fourier 181 Coefficient(s), Fourier — Walsh 182 Complex form of Fourier series 201 Complex number(s) 9 Complex number(s) algebraic form of 10 Complex number(s) argument of 11 Complex number(s) conjugate 10 Complex number(s) difference of 11 Complex number(s) equal 10 Complex number(s) exponential form of 12 Complex number(s) imaginary part of 9 Complex number(s) modulus of 11 Complex number(s) natural logarithm of 28 Complex number(s) nth root of 14 Complex number(s) principal value of 12 Complex number(s) quotient of 11 Complex number(s) real part of 9 Complex number(s) sum of 10 Complex number(s) trigonometric form of 12 Convolution of two functions 309 Curve(s), piecewise smooth 17 Curve(s), smooth 17 Differential equation(s), Bessel’s 156 Differential equation(s), general solution of 159—160 Differential equation(s), integration of 153 Differential equation(s), Legendre’s 173 Differential equation(s), linear 154 Domain(s) of values of a function 22 Domain(s), closed 16 Domain(s), multiply connected 17 Domain(s), simply connected 17 Domain(s), singly connected 17 Equation(s) heat 241 Equation(s) heat-conduction 330—335 Equation(s) of a long line 336 Equation(s) of diffusion in an impenetrable tube 332—335 Equation(s) of telegraphy 336 Equation(s) telegrapher’s 335—336 Forced vibrations of a fixed string 234 Formula (s), Cauchy — Hadamard 121 Formula (s), Cauchy’s integral 69 Formula (s), Christoffel — Darboux 179 Formula (s), De Moivre 14 Formula (s), Euler’s 12 Formula (s), Green’s 64—65 Formula (s), Rodrigues 174 Fourier’s double series 211—216 Free vibrations of a string 230—234 Function(s) Bessel’s 158—160 Function(s) binomial 129 Function(s) derivative of 29 Function(s) differentiation of 29—38 Function(s) Dirac 252 Function(s) elementary 27 Function(s) exponential 127 Function(s) geometrical meaning of 23 Function(s) limit of 25—26 Function(s) linear 42 Function(s) linear-fractional 44—47 Function(s) logarithmic 28 55 130 Function(s) many-valued 22 Function(s) many-valued, branch points of 25 Function(s) one-sheeted 25 Function(s) one-valued 22 Function(s) original 297 Function(s) orthogonal systems of 162 Function(s) piecewise continuous 192 Function(s) power 48 Function(s) single-valued 22 Function(s) spectral 224 Function(s) trigonometric 57 128 Function(s) uniformly continuous in a domain 27 Function(s) value of 22 Function(s) Zhukovsky’s 50 Function(s) Zhukovsky’s inverse of 54 Function(s), of a complex variable 22 Function(s), of a complex variable, analytic in a domain 29 Function(s), of a complex variable, analytic in a domain, properties of 34—36 Geometrical meaning of a function of a complex variable 23 Geometrical meaning of the argument of a derivative 36 Geometrical meaning of the modulus of a derivative 38 Heat conduction 241 Heat conduction in a homogeneous cylinder 245 Heat conduction in an infinite rod 241 Heat equation 241 Heat-conduction equation 330—335 Inequality, Bessel’s 185 Inequality, Buniakovski — Cauchy 179 Inequality, Schwarz’s 179 Integral(s) Dirichlet 194 Integral(s) Duhamel’s 311—313 Integral(s) of the Cauchy type 76—78 Integral(s) with respect to a complex variable 61—69 Integral(s), Fourier’s 216—222 Integral(s), Fourier’s in complex form 220 Integral(s), Fourier’s in real form 221 Integral(s), Fourier’s spectral characteristics of 223 Integral(s), indefinite, in a complex domain 66 Integral(s), Mellin's 319—325 Integration of differential equations 153 Inverse Laplace transformation 314—319 inversion 42 Jordan’s lemma 282 Kernel, Dirichlet’s 194 215 Lemma(s), Jordan’s 282 Lemma(s), Riemann — Lebesgue 192 Limit of a number sequence 18 Linearity 301 Logarithmic derivative 288 Mapping(s), conformal at a point 38 Mapping(s), conformal in a domain 39 Mapping(s), determined by linear and linear-fractional functions 42—47 Mapping(s), linear-fractional 44 Mapping(s), properties of 39—41 Mapping(s), superposition of 23 Maximum Modulus Principle 72—74 Method(s), artificial, for expanding functions into Taylor’s series 131 Method(s), Euler’s 150 Method(s), Maliev’s 207—211 Method(s), operational 297 Method(s), Rummer’s 147 Modulus of derivative 38 Modulus of derivative, geometrical meaning of 38 Modulus of the product of two complex numbers 13 Numerical series 79 Orthogonal expansions 180—182 Point(s), boundary, of a domain 16 Point(s), essential singular 268 Point(s), interior, of a domain 16 Point(s), regular 139 Point(s), removable singular 268 Point(s), singular 139 Polynomial(s), Chebyshev 175—177 Polynomial(s), Jacobi 177—179 Polynomial(s), Legendre’s 173—174 Polynomial(s), orthogonal 173—179 Polynomial(s), ultraspherical 179 primitive 68 Properties of analytic functions 72—74 Properties of discrete Fourier’s transformations 249—257 Quotient of two complex numbers 11 R-neighbourhood of the point at infinity 15 Residue(s) 275—279 Residue(s) definition of 275—277 Residue(s) evaluation of 255—277 Residue(s) logarithmic 288 Residue(s) of an analytic function 275 Riemann surfaces 138 Series, absolute convergence of 85 Series, alternating 99 Series, application of 142—146 Series, binomial 120 Series, conditional convergence of 85 Series, convergent 79 Series, convergent, properties of 82—85 Series, Fourier 181 Series, Fourier of nonperiodic functions 203 Series, Fourier spectral characteristics of 223 Series, Fourier spectral density of 224 Series, Fourier trigonometric 182 189—192 Series, Fourier — Bessel 182 Series, Fourier, complex form of 201—202 Series, general functional 103—110 Series, harmonic 81 Series, Laurent’s 264 Series, Laurent’s numerical 79 Series, Laurent’s orthogonal 179—188 Series, Laurent’s partial sums of 79 Series, Laurent’s principal part of 265 Series, Laurent’s regular part of 265 Series, Laurent’s sum of 79 Series, Laurent’s Taylor’s 124 Series, uniform convergence of 106 Series, uniform convergence of Cauchy’s criterion for 106 Series, uniform convergence of conditions of 107—110 Series, uniform convergence of Weierstrass’ test for 107— 108 Series, with arbitrary terms 100 Singularity, essential 268 Singularity, isolated 268 Singularity, removable 268 Spectrum 224 228 Spectrum, amplitude 224 228 Spectrum, continuous 252 Spectrum, discrete 252 Spectrum, phase 224 228 Sturm — Liouville problem 168 Sturm — Liouville problem, eigenfunction of 171 Sturm — Liouville problem, eigenvalue of 171 System(s) of Bessel functions 164 System(s) of eigenfunctions of Sturm — Liouville problem 168 System(s) of Raaemacher functions,[ 166 System(s) of Walsh functions 167 System(s) orthogonal 162 System(s) trigonometric 163 Test(s), Cauchy’s 93 Test(s), Cauchy’s integral 95—98 Test(s), comparison 89 Test(s), Dirichlet — Abel 100 Test(s), D’Alembert’s 91 Test(s), Leibniz 99 Test(s), limit comparison 90 Test(s), necessary, for convergence of a series 81 Test(s), second Weierstrass’, for uniform convergence 109 Test(s), sufficient, for absolute convergence of numerical series 89—98 Test(s), Weierstrass’, for uniform convergence 107 Theorem(s) on a logarithmic residue 288 Theorem(s) on convergence of Fourier series 195—199 Theorem(s) on uniform convergence 198 Theorem(s) on uniqueness of analytic functions 134 Theorem(s), Borel s 310 Theorem(s), Cantor’s 27 Theorem(s), Cauchy’s 64 Theorem(s), Cauchy’s integral 65 Theorem(s), Laurent’s 266 Theorem(s), Liouville’s 78 Theorem(s), mean-value 72 Theorem(s), Morera’s 78 Theorem(s), power series 122—123 Theorem(s), residue 278 Theorem(s), Riemann mapping 41 Theorem(s), Riemann’s 88 Theorem(s), RouchS’s 292 Theorem(s), Sochozky’s 271 Theorem(s), Taylor’s 124 Theorem(s), uniqueness 136 238 Transform(s) 298 Transform(s) of some elementary functions 300—301 Transform(s), differentiation of 306 Transform(s), Fourier’s convolution of 250 Transform(s), Fourier’s cosine 221 Transform(s), Fourier’s fast 249 Transform(s), Fourier’s in complex form 222 Transform(s), Fourier’s in real form 221 Transform(s), Fourier’s sine 221 Transform(s), integration of 307 Transform(s), Laplace 297 Transform(s), properties of 301—309 Transformation(s), Abel’s 100 Transformation(s), Fourier’s 200 Transformation(s), Fourier’s discrete 249 Transformation(s), Heaviside 299 Two-sheeted Riemann surface 25 Unit imaginary number 9 10 Variable, dependent 22 Variable, independent 22 Weierstrass’ test for uniform convergence 107—108 Zhukovsky’s function 50
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