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Íàçâàíèå: Mathematical Analysis (Advanced Topics). Part 1. General Functional Series and Their Applications
Àâòîð: 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...
Acceleration oj series convergence146—152 Algorithm of fast Fourier’s transform257—263 Analytic continuation137 Argument22 Argument of a complex number11 Argument of a derivative36 Argument of a derivative, geometrical meaning of36 Argument of the product of two complex numbers13 Argument of the quotient of two complex numbers13 Basic frequency203 Boundary correspondence principle41 Cauchy — Riemann conditions30 Cauchy’s criterion81103 Cauchy’s integral and its application69—78 Circle of convergence120 Coefficient(s), Fourier181 Coefficient(s), Fourier — Walsh182 Complex form of Fourier series201 Complex number(s)9 Complex number(s) algebraic form of10 Complex number(s) argument of11 Complex number(s) conjugate10 Complex number(s) difference of11 Complex number(s) equal10 Complex number(s) exponential form of12 Complex number(s) imaginary part of9 Complex number(s) modulus of11 Complex number(s) natural logarithm of28 Complex number(s) nth root of14 Complex number(s) principal value of12 Complex number(s) quotient of11 Complex number(s) real part of9 Complex number(s) sum of10 Complex number(s) trigonometric form of12 Convolution of two functions309 Curve(s), piecewise smooth17 Curve(s), smooth17 Differential equation(s), Bessel’s156 Differential equation(s), general solution of159—160 Differential equation(s), integration of153 Differential equation(s), Legendre’s173 Differential equation(s), linear154 Domain(s) of values of a function22 Domain(s), closed16 Domain(s), multiply connected17 Domain(s), simply connected17 Domain(s), singly connected17 Equation(s) heat241 Equation(s) heat-conduction330—335 Equation(s) of a long line336 Equation(s) of diffusion in an impenetrable tube332—335 Equation(s) of telegraphy336 Equation(s) telegrapher’s335—336 Forced vibrations of a fixed string234 Formula (s), Cauchy — Hadamard121 Formula (s), Cauchy’s integral69 Formula (s), Christoffel — Darboux179 Formula (s), De Moivre14 Formula (s), Euler’s12 Formula (s), Green’s64—65 Formula (s), Rodrigues174 Fourier’s double series211—216 Free vibrations of a string230—234 Function(s) Bessel’s158—160 Function(s) binomial129 Function(s) derivative of29 Function(s) differentiation of29—38 Function(s) Dirac 252 Function(s) elementary27 Function(s) exponential127 Function(s) geometrical meaning of23 Function(s) limit of25—26 Function(s) linear42 Function(s) linear-fractional44—47 Function(s) logarithmic2855130 Function(s) many-valued22 Function(s) many-valued, branch points of25 Function(s) one-sheeted25 Function(s) one-valued22 Function(s) original297 Function(s) orthogonal systems of162 Function(s) piecewise continuous192 Function(s) power48 Function(s) single-valued22 Function(s) spectral224 Function(s) trigonometric57128 Function(s) uniformly continuous in a domain27 Function(s) value of22 Function(s) Zhukovsky’s50 Function(s) Zhukovsky’s inverse of54 Function(s), of a complex variable22 Function(s), of a complex variable, analytic in a domain29 Function(s), of a complex variable, analytic in a domain, properties of34—36 Geometrical meaning of a function of a complex variable23 Geometrical meaning of the argument of a derivative36 Geometrical meaning of the modulus of a derivative38 Heat conduction241 Heat conduction in a homogeneous cylinder245 Heat conduction in an infinite rod241 Heat equation241 Heat-conduction equation330—335 Inequality, Bessel’s185 Inequality, Buniakovski — Cauchy179 Inequality, Schwarz’s179 Integral(s) Dirichlet194 Integral(s) Duhamel’s311—313 Integral(s) of the Cauchy type76—78 Integral(s) with respect to a complex variable61—69 Integral(s), Fourier’s216—222 Integral(s), Fourier’s in complex form220 Integral(s), Fourier’s in real form221 Integral(s), Fourier’s spectral characteristics of223 Integral(s), indefinite, in a complex domain66 Integral(s), Mellin's319—325 Integration of differential equations153 Inverse Laplace transformation314—319 inversion42 Jordan’s lemma282 Kernel, Dirichlet’s194215 Lemma(s), Jordan’s282 Lemma(s), Riemann — Lebesgue192 Limit of a number sequence18 Linearity301 Logarithmic derivative288 Mapping(s), conformal at a point38 Mapping(s), conformal in a domain39 Mapping(s), determined by linear and linear-fractional functions42—47 Mapping(s), linear-fractional44 Mapping(s), properties of39—41 Mapping(s), superposition of23 Maximum Modulus Principle72—74