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Efimov A.V. Mathematical Analysis (Advanced Topics). Part 1. General Functional Series and Their Applications
Efimov A.V.  Mathematical Analysis (Advanced Topics). Part 1. General Functional Series and Their Applications







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: Mathematical Analysis (Advanced Topics). Part 1. General Functional Series and Their Applications

: Efimov A.V.

:

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...


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: 1985

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: 12.10.2013

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Acceleration oj series convergence      146152
Algorithm of fast Fouriers transform      257263
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
Cauchys criterion      81 103
Cauchys integral and its application      6978
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), Bessels      156
Differential equation(s), general solution of      159160
Differential equation(s), integration of      153
Differential equation(s), Legendres      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      330335
Equation(s) of a long line      336
Equation(s) of diffusion in an impenetrable tube      332335
Equation(s) of telegraphy      336
Equation(s) telegraphers      335336
Forced vibrations of a fixed string      234
Formula (s), Cauchy Hadamard      121
Formula (s), Cauchys integral      69
Formula (s), Christoffel Darboux      179
Formula (s), De Moivre      14
Formula (s), Eulers      12
Formula (s), Greens      6465
Formula (s), Rodrigues      174
Fouriers double series      211216
Free vibrations of a string      230234
Function(s) Bessels      158160
Function(s) binomial      129
Function(s) derivative of      29
Function(s) differentiation of      2938
Function(s) Dirac $\delta -$      252
Function(s) elementary      27
Function(s) exponential      127
Function(s) geometrical meaning of      23
Function(s) limit of      2526
Function(s) linear      42
Function(s) linear-fractional      4447
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) Zhukovskys      50
Function(s) Zhukovskys 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      3436
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      330335
Inequality, Bessels      185
Inequality, Buniakovski Cauchy      179
Inequality, Schwarzs      179
Integral(s) Dirichlet      194
Integral(s) Duhamels      311313
Integral(s) of the Cauchy type      7678
Integral(s) with respect to a complex variable      6169
Integral(s), Fouriers      216222
Integral(s), Fouriers in complex form      220
Integral(s), Fouriers in real form      221
Integral(s), Fouriers spectral characteristics of      223
Integral(s), indefinite, in a complex domain      66
Integral(s), Mellin's      319325
Integration of differential equations      153
Inverse Laplace transformation      314319
inversion      42
Jordans lemma      282
Kernel, Dirichlets      194 215
Lemma(s), Jordans      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      4247
Mapping(s), linear-fractional      44
Mapping(s), properties of      3941
Mapping(s), superposition of      23
Maximum Modulus Principle      7274
Method(s), artificial, for expanding functions into Taylors series      131
Method(s), Eulers      150
Method(s), Malievs      207211
Method(s), operational      297
Method(s), Rummers      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      180182
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      175177
Polynomial(s), Jacobi      177179
Polynomial(s), Legendres      173174
Polynomial(s), orthogonal      173179
Polynomial(s), ultraspherical      179
primitive      68
Properties of analytic functions      7274
Properties of discrete Fouriers transformations      249257
Quotient of two complex numbers      11
R-neighbourhood of the point at infinity      15
Residue(s)      275279
Residue(s) definition of      275277
Residue(s) evaluation of      255277
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      142146
Series, binomial      120
Series, conditional convergence of      85
Series, convergent      79
Series, convergent, properties of      8285
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 189192
Series, Fourier Bessel      182
Series, Fourier, complex form of      201202
Series, general functional      103110
Series, harmonic      81
Series, Laurents      264
Series, Laurents numerical      79
Series, Laurents orthogonal      179188
Series, Laurents partial sums of      79
Series, Laurents principal part of      265
Series, Laurents regular part of      265
Series, Laurents sum of      79
Series, Laurents Taylors      124
Series, uniform convergence of      106
Series, uniform convergence of Cauchys criterion for      106
Series, uniform convergence of conditions of      107110
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), Cauchys      93
Test(s), Cauchys integral      9598
Test(s), comparison      89
Test(s), Dirichlet Abel      100
Test(s), DAlemberts      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      8998
Test(s), Weierstrass, for uniform convergence      107
Theorem(s) on a logarithmic residue      288
Theorem(s) on convergence of Fourier series      195199
Theorem(s) on uniform convergence      198
Theorem(s) on uniqueness of analytic functions      134
Theorem(s), Borel s      310
Theorem(s), Cantors      27
Theorem(s), Cauchys      64
Theorem(s), Cauchys integral      65
Theorem(s), Laurents      266
Theorem(s), Liouvilles      78
Theorem(s), mean-value      72
Theorem(s), Moreras      78
Theorem(s), power series      122123
Theorem(s), residue      278
Theorem(s), Riemann mapping      41
Theorem(s), Riemanns      88
Theorem(s), RouchSs      292
Theorem(s), Sochozkys      271
Theorem(s), Taylors      124
Theorem(s), uniqueness      136 238
Transform(s)      298
Transform(s) of some elementary functions      300301
Transform(s), differentiation of      306
Transform(s), Fouriers convolution of      250
Transform(s), Fouriers cosine      221
Transform(s), Fouriers fast      249
Transform(s), Fouriers in complex form      222
Transform(s), Fouriers in real form      221
Transform(s), Fouriers sine      221
Transform(s), integration of      307
Transform(s), Laplace      297
Transform(s), properties of      301309
Transformation(s), Abels      100
Transformation(s), Fouriers      200
Transformation(s), Fouriers 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      107108
Zhukovskys function      50
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