...s of a complex variable, theory of|Functions of a complex variable, theory of]]).
423 bytes (68 words) - 16:59, 7 February 2011
''simple function (of a complex variable)''
...older, names for a [[Univalent function|univalent function]] of a complex variable.
659 bytes (91 words) - 17:05, 7 February 2011
''in the theory of functions of a complex variable''
If $f(z)$ is a regular analytic function in the disc $|z|<R$, $\alpha$ is a positive number, and if
2 KB (260 words) - 17:48, 1 November 2014
''in a domain''
...tion at a point $a$ is a function that is regular in some neighborhood of $a$.
537 bytes (80 words) - 21:41, 17 October 2014
...ons]] $\tan z$ and $\operatorname{cotan}z$ are meromorphic simply-periodic functions with fundamental period $2\omega=\pi$.
...riod $2\omega\in X$. A non-constant continuous periodic function of a real variable is necessarily simply-periodic.
897 bytes (128 words) - 18:55, 17 October 2014
$#A+1 = 24 n = 0
$#C+1 = 24 : ~/encyclopedia/old_files/data/I052/I.0502370 Inverse hyperbolic functions
3 KB (487 words) - 19:15, 17 January 2024
...ww.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264201.png" />,
of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofm
...//www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642011.png" />
of the domain <img align="absmiddle" border="0" src="https://www.encyclopediao
4 KB (623 words) - 17:10, 7 February 2011
$#A+1 = 24 n = 0
If a function
3 KB (454 words) - 08:12, 6 June 2020
$#A+1 = 19 n = 0
A function of the form
2 KB (328 words) - 19:37, 5 June 2020
...d on considering all possible elements of an analytic function in the form of generalized power series ([[Puiseux series]])
...table topology, the analytic image is converted to the [[Riemann surface]] of the given function.
1 KB (199 words) - 17:44, 1 November 2014
...ng all paths in an arbitrary domain $D\subset\mathbf C^n$, then the result of this continuation at any point $z^*\in D$ (that is, the element $\Sigma(z^*
The monodromy theorem is valid also for analytic functions $f(z)$ defined in domains $D$ on Riemann surfaces or on Riemann domains. Se
2 KB (315 words) - 10:59, 27 June 2015
$#A+1 = 12 n = 0
$#C+1 = 12 : ~/encyclopedia/old_files/data/E036/E.0306950 Extended complex plane
2 KB (233 words) - 19:38, 5 June 2020
...is an extremal function in a number of problems in the theory of univalent functions (cf. [[Bieberbach conjecture]]; [[Univalent function]]).
...]</TD> <TD valign="top"> W.K. Hayman, "Coefficient problems for univalent functions and related function classes" ''J. London Math. Soc.'' , '''40''' : 3 (1
1 KB (188 words) - 19:36, 5 April 2023
''in the theory of functions of a complex variable''
...t is necessary and sufficient that it be uniformly bounded in the interior of $D$, that is, uniformly bounded on any compactum $K\subset D$.
2 KB (306 words) - 13:06, 28 August 2014
...omorphic functions $f(z)$ in the space $\mathbf C^n$, $n\geq2$, of several complex variables $z=(z_1,\ldots,z_n)$.
...ranscendental function is the presence of either a singularity that is not a pole or an algebraic branch point; e.g. the logarithmic function $\ln z$ ha
3 KB (477 words) - 13:50, 9 April 2023
...phic function|holomorphic functions]] of one complex variable. In the case of the unit circle one has the following relationship between the Cauchy kerne
See also [[Kernel of an integral operator]].
1 KB (198 words) - 20:26, 18 March 2024
...of functions of a complex variable|Approximation of functions of a complex variable]]).
...ompact set $K\subset D$ and $\epsilon>0$ there is a polynomial $p(z)$ with complex coefficients such that $|f(z)-p(z)|<\epsilon$ for all $z\in K$.
3 KB (531 words) - 18:58, 1 May 2014
''conformal mapping of the second kind''
...gles between the curves passing through $z_0$ but changes the orientation. A function $f(z)$ which produces an anti-conformal mapping is an [[Anti-holom
705 bytes (104 words) - 15:14, 17 July 2014
''for a meromorphic function $f(z)$ in a domain $G$ of the complex $z$-plane''
...inside $D$ within any angle with vertex $\zeta$ formed by a pair of chords of $D$.
2 KB (264 words) - 21:15, 1 November 2014
...$ be valid; then in the domain $G$ the sum $f(z)+g(z)$ has the same number of zeros as $f(z)$.
...f the|Argument, principle of the]]) and it implies the fundamental theorem of algebra for polynomials.
3 KB (461 words) - 12:23, 14 February 2020
$#A+1 = 37 n = 0
Montel's theorem on the approximation of analytic functions by polynomials: If $ D $
3 KB (452 words) - 12:08, 18 February 2022
''on conformal mapping of domains with variable boundaries''
...$D_{z_0}$ is assumed to be bounded by a Jordan curve. Then the sequence of functions $\{w=f_n(z)\}$ which map $D_n$ conformally onto the disc $\Delta=\{w:|w|<1\
1 KB (209 words) - 21:14, 22 December 2018
''exact radius of star-likeness, bound of star-likeness''
...y number $r$ in the interval $0<r<R_U$ is called a radius of star-likeness of the class $U$.
1 KB (245 words) - 08:26, 28 April 2023
''on uniform approximation by entire functions''
...asing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ such
2 KB (243 words) - 18:27, 13 August 2015
...c continuation]] of a given element of an analytic function represented by a power series
\Pi(a;r) = \sum_{\nu=0}^\infty c_\nu (z-a)^\nu
2 KB (238 words) - 21:25, 13 December 2016
...on of the behaviour of specific functions. For the relief of the principal functions see [[#References|[2]]].
...align="top">[2]</TD> <TD valign="top"> E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German
930 bytes (137 words) - 11:16, 5 October 2014
$#A+1 = 44 n = 0
$#C+1 = 44 : ~/encyclopedia/old_files/data/C025/C.0205040 Conjugate harmonic functions,
4 KB (617 words) - 17:46, 4 June 2020
...n|analytic functions]] of one complex variable and sometimes in the theory of [[Partial differential equation|partial differential equations]].
...$ is called a ''removable singular point'' of the function $f$ if there is a positive radius $r_0$ such that $f$ is bounded on the punctured disk $\{z:
2 KB (344 words) - 08:39, 15 January 2014
One of the [[Trigonometric functions|trigonometric functions]]:
Its domain of definition is the entire real line; its range of values is the closed interval $[-1,1]$; the cosine is an even periodic func
2 KB (283 words) - 12:43, 14 February 2020
...w D$, just as for every analytic function of one complex variable there is a corresponding [[Riemann surface]]; the function $f$ is single-valued on $\t
...D valign="top">[1]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1976) (In Russian)</TD></TR>
1 KB (205 words) - 17:02, 13 June 2020
...bharmonic function]]). For $n=1$ they correspond to logarithmically-convex functions.
...ve linear combination, of several logarithmically-subharmonic functions is a logarithmically-subharmonic function.
1 KB (205 words) - 15:22, 29 December 2018
$#A+1 = 40 n = 0
$#C+1 = 40 : ~/encyclopedia/old_files/data/A012/A.0102210 Analytic curve,
3 KB (480 words) - 18:31, 5 April 2020
$#A+1 = 52 n = 0
''for an element of an analytic function $ f ( z) $''
5 KB (764 words) - 22:13, 5 June 2020
''Riemann domain, complex (-analytic) manifold over $ \mathbf C ^{n} $''
An analogue of the [[Riemann surface|Riemann surface]] of an analytic function $ w = f(z) $
4 KB (633 words) - 19:58, 28 January 2020
...als. Let $K$ be a compact subset of the complex $z$-plane $\mathbf C$ with a connected complement. Then every function $f$ continuous on $K$ and holomor
...eory in the complex plane and has many applications in various branches of complex analysis.
5 KB (707 words) - 19:02, 1 May 2014
...www.encyclopediaofmath.org/legacyimages/i/i051/i051910/i0519105.png" />
of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...inside the respective domain) are also valid for other classes
of analytic
functions.
6 KB (915 words) - 17:24, 7 February 2011
Out of 36 formulas, 36 were replaced by TEX code.-->
...e } C ( X )$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [[#References|[a4]]]
3 KB (474 words) - 19:56, 27 February 2021
The classical polylogarithms are special functions of a complex variable $z$, defined by the series
...umber inside the unit disk and $n$ an integer at least equal to $1$. These functions satisfy the differential equations
2 KB (237 words) - 19:07, 1 May 2023
$#A+1 = 52 n = 0
corresponding to the value of the argument $ x $,
6 KB (901 words) - 12:10, 1 May 2023
$#A+1 = 75 n = 0
...percomplex function theory (cf. also [[Hypercomplex functions|Hypercomplex functions]]), renamed Clifford analysis in the 1980s, when it grew into an autonomous
9 KB (1,297 words) - 17:44, 4 June 2020
$#A+1 = 4 n = 0
of a complex variable $ z $,
2 KB (240 words) - 08:18, 6 January 2024
$#A+1 = 30 n = 0
''branch point of infinite order''
3 KB (399 words) - 22:17, 5 June 2020
$#A+1 = 117 n = 0
$#C+1 = 117 : ~/encyclopedia/old_files/data/T094/T.0904210 Trigonometric functions
12 KB (1,667 words) - 08:26, 6 June 2020
The formula expressing the rule for raising a [[Complex number|complex number]], expressed in trigonometric form
...n $n$-th power. According to de Moivre's formula the modulus $\rho$ of the complex number is raised to that power and the argument $\varphi$ is multiplied by
1 KB (241 words) - 10:03, 4 June 2013
$#A+1 = 71 n = 0
''of a Green function''
6 KB (963 words) - 14:03, 18 May 2023
$#A+1 = 29 n = 0
A single-valued analytic function $ f ( z) $
4 KB (565 words) - 19:36, 5 June 2020
$#A+1 = 18 n = 0
of the domain of definition $ D $
2 KB (247 words) - 08:03, 6 June 2020
...arzian differential parameter, of an analytic function $f(z)$ of a complex variable $z$''
...mal mapping]] of polygons onto the disc, in particular in the studies of H.A. Schwarz [[#References|[1]]].
3 KB (443 words) - 11:10, 30 December 2018
$#A+1 = 14 n = 0
The value of a function $ f ( z) $,
2 KB (301 words) - 08:09, 6 June 2020
...Abel in {{Cite|Ab}}. If $a_1, \ldots, a_N$, $b_1, \ldots, b_N$, are given complex numbers and we set
Observe that, if $B_0$ is arbitrarily chosen and we modify the definition of $B_n$ by
2 KB (382 words) - 13:06, 10 December 2013
...one of its sheets a largest open disc of radius $B_f > 0$. It was shown by A. Bloch [[#References|[1]]] that
A precise estimate for $B$ is
2 KB (312 words) - 17:31, 13 November 2016
...ve terms the value of which is not less than the values of all other terms of this series.
Applying this idea to the study of [[Power series|power series]]
1 KB (256 words) - 13:07, 12 December 2013
$#A+1 = 91 n = 0
$#C+1 = 91 : ~/encyclopedia/old_files/data/A010/A.0100460 Abstract analytic function,
6 KB (901 words) - 16:08, 1 April 2020
...a is sometimes used also for a useful property of sequences of subarhominc functions.
...\mathbb C$ is holomorphic at every point $\zeta\in D$ with respect to each variable $z_k$, then it is holomorphic with respect to all variables.
6 KB (1,001 words) - 18:22, 31 March 2017
...Ref|Ca}}||valign="top"| C. Caratheodory, "Theory of functions of a complex variable", '''1''', Chelsea (1983) Sects. 274–275
662 bytes (94 words) - 21:40, 27 April 2012
...arities]] of [[Holomorphic function|holomorphic functions]] of one complex variable
...o \mathbb C$ be an holomorphic function and $z_0$ a point for which $U$ is a punctured neighborhood. Then either the limit
3 KB (465 words) - 20:00, 14 December 2014
...n there exists at least one ray $S=\{z:\arg(z-a)=\theta_0\}$ issuing from $a$ such that in every angle
$$V=\{z:|\arg(z-a)-\theta_0|<\epsilon\},\quad\epsilon>0,$$
2 KB (343 words) - 21:49, 22 December 2018
...$-\infty\leqslant\alpha<\sigma<\beta\leqslant+\infty$, and expandable into a series
...\tau$ is called an $\varepsilon$-almost-period of $f(s)$ if for all points of the strip $(\alpha, \beta)$ the inequality
2 KB (348 words) - 06:21, 16 October 2013
...es was published by B. Taylor in 1715, whereas a series reducible to it by a simple transformation was published by Johann I. Bernoulli in 1694.
===One real variable===
4 KB (710 words) - 06:13, 13 June 2022
$#A+1 = 7 n = 0
A common singular point $ z _ {0} $
846 bytes (120 words) - 19:39, 5 June 2020
...|z|<1$ onto a [[Star-like domain|star-like domain]] with respect to $w=0$. A function $f(z)$, $f(z)\neq0$ in $0<|z|<1$, $f(0)=0$, $f'(0)\neq0$, regular
...ormalized so that $f(0)=0$, $f'(0)=1$, forms the class $S^*$, which admits a parametric representation by Stieltjes integrals:
3 KB (519 words) - 14:43, 30 July 2014
Formulas connecting the exponential and trigonometric functions:
These hold for all values of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become
1 KB (171 words) - 12:50, 10 August 2014
$#A+1 = 1 n = 0
A mapping of one topological space into another under which the image of every open set is itself open.
3 KB (511 words) - 08:04, 6 June 2020
$#A+1 = 62 n = 0
A complex variable $ t $
5 KB (620 words) - 12:06, 9 December 2023
$#A+1 = 52 n = 0
...ent analytic functions in a complex domain. Suppose that the complement of a bounded continuum $ K $
5 KB (711 words) - 19:38, 5 June 2020
$#A+1 = 18 n = 0
$#C+1 = 18 : ~/encyclopedia/old_files/data/A012/A.0102510 Angular boundary value,
2 KB (293 words) - 18:47, 5 April 2020
...aginary parts of a complex-valued function $f$ of one (or several) complex variable so that $f$ is [[Holomorphic function|holomorphic]].
===One complex variable===
7 KB (1,063 words) - 12:51, 22 March 2023
...st partial fractions. Let $\{a_n\}_{n=1}^\infty$ be a sequence of distinct complex numbers,
and let $\{g_n(z)\}$ be a sequence of rational functions of the form
6 KB (980 words) - 18:47, 24 May 2017
...r integrals (cf. [[Fourier integral|Fourier integral]]), which are objects of harmonic analysis.
2 KB (287 words) - 11:59, 27 September 2012
$#A+1 = 25 n = 0
.../encyclopedia/old_files/data/B015/B.0105820 Besicovitch almost\AAhperiodic functions
4 KB (506 words) - 14:27, 30 May 2020
An expression of the form
...said to be ''convergent'' if there exists a non-zero limit of the sequence of partial products
5 KB (741 words) - 17:20, 24 April 2016
...e $\Gamma$ the same number of zeros as $f(z)$ inside $\Gamma$. Obtained by A. Hurwitz .
<TR><TD valign="top">[1a]</TD> <TD valign="top"> A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würze
1 KB (221 words) - 18:17, 3 January 2015
$#A+1 = 61 n = 0
A function having a period (cf. [[Period of a function|Period of a function]]).
5 KB (793 words) - 08:05, 6 June 2020
...ss property of analytic functions (cf. [[Uniqueness properties of analytic functions]]) (see {{Cite|LuPr}}).
...in $D$ that has infinite angular boundary values on a set $E\subset\Gamma$ of positive measure.
3 KB (424 words) - 21:56, 24 July 2012
$#A+1 = 42 n = 0
...: ~/encyclopedia/old_files/data/A012/A.0102250 Analytic function, element of an
4 KB (679 words) - 18:34, 5 April 2020
''with respect to a fixed point $O$''
...pace $\mathbf C^n$, $n\geq1$, such that, for any point of $D$, the segment of the straight line from that point to $O$ lies entirely in $D$.
2 KB (305 words) - 14:28, 30 July 2014
$#A+1 = 75 n = 0
is the class of functions $ f ( z) = z + \dots $
7 KB (959 words) - 22:14, 5 June 2020
$#A+1 = 36 n = 0
$#C+1 = 36 : ~/encyclopedia/old_files/data/D032/D.0302970 Disc of convergence
4 KB (602 words) - 19:35, 5 June 2020
...omplex (independent) variables $z=x+iy$, $z^*=x-iy$ are introduced instead of the coordinates $x,y$ in the plane; note that the Laplace potential operato
Then the transform is realized by a complex line integral (cf. [[Curvilinear integral|Curvilinear integral]]) as
4 KB (623 words) - 15:18, 14 February 2020
$#A+1 = 24 n = 0
...on to the [[Coefficient problem|coefficient problem]] for bounded analytic functions. They were obtained by I. Schur [[#References|[1]]]. Let $ B $
4 KB (452 words) - 00:51, 21 January 2022
...domain is representable as the kernel of a decreasing convergent sequence of simply-connected domains $\{ G_n\}$:
and every domain $G$ for which there exists such a sequence is a Carathéodory domain (Carathéodory's theorem, see {{Cite|Ca}}).
2 KB (269 words) - 10:07, 22 April 2012
''of an analytic function''
...ular points (cf. [[Singular point]]) of an analytic function $f(z)$ in the complex variables $z=(z_1,\ldots,z_n)$, $n\geq1$, defined by some supplementary con
1 KB (198 words) - 18:51, 25 October 2014
$#A+1 = 73 n = 0
...edia/old_files/data/F042/F.0402120 Functions of a complex variable, theory of
13 KB (1,922 words) - 19:55, 1 February 2022
$#A+1 = 31 n = 1
A power series of the form
3 KB (470 words) - 08:17, 26 March 2023
$#A+1 = 79 n = 0
$#C+1 = 79 : ~/encyclopedia/old_files/data/C024/C.0204800 Conformal radius of a domain
5 KB (759 words) - 16:51, 5 June 2020
...irst proved by G. Morera in {{Cite|Mo}}, which is an (incomplete) converse of the [[Cauchy integral theorem]]. The theorem states the following.
Let $D\subset \mathbb C$ be an open set and $f: D\to \mathbb C$ a continuous function. If
2 KB (358 words) - 17:23, 12 January 2014
$#A+1 = 155 n = 1
''of a compact set''
12 KB (1,769 words) - 14:55, 30 June 2020
Power series in one complex variable $ z $.
A series (representing a function) of the form
16 KB (2,404 words) - 13:34, 4 November 2023
...ower of a given number $r$ (the radius of the lemniscate). The equation of a lemniscate in rectangular Cartesian coordinates is
...cle is a lemniscate with one focus, and a [[Cassini oval|Cassini oval]] is a lemniscate with two foci. See also [[Bernoulli lemniscate|Bernoulli lemnisc
2 KB (304 words) - 06:49, 7 April 2023
$#A+1 = 27 n = 0
is a complex summable function on a closed rectifiable Jordan curve $ L $
3 KB (499 words) - 13:59, 30 April 2023
...card theorem]] on the behaviour of an analytic function in a neighbourhood of an essential singularity.
Iversen's theorem has been extended to subharmonic functions on $\R^n$, notably by W.K. Hayman, see {{Cite|HaKe}}, {{Cite|Ha}}.
1 KB (198 words) - 21:07, 31 July 2012
$#A+1 = 32 n = 1
be a holomorphic function on the disc $ E = \{ | z | < 1 \} $,
5 KB (742 words) - 08:12, 6 June 2020
A basic concept in mathematical analysis.
...opediaofmath.org/legacyimages/c/c025/c025650/c0256509.png" /> there exists
a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
26 KB (3,622 words) - 17:12, 7 February 2011
...Riemann surface — its double (cf. [[Double of a Riemann surface|Double of a Riemann surface]]).
...p">[1]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)</TD></TR>
1 KB (159 words) - 10:51, 18 October 2014
where $x$ is a complex variable and $\nu$ a complex parameter, first studied by Weber in {{Cite|We}}. The term Weber function
...econd type $Y_\nu$ (also called Bessel functions of second type or Neumann functions and denoted sometimes by $N_\nu$).
2 KB (367 words) - 09:12, 22 February 2014
$#A+1 = 11 n = 0
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], denoted by
2 KB (241 words) - 17:31, 5 June 2020
$#A+1 = 31 n = 0
$#C+1 = 31 : ~/encyclopedia/old_files/data/A013/A.0103750 Asymptotic power series
4 KB (542 words) - 06:35, 14 April 2024
...modulus principle is that, under the same conditions as above, the modulus of $f(z)$ does not attain its least upper bound
...ytic) manifold, in particular, on a [[Riemann surface|Riemann surface]] or a [[Riemannian domain|Riemann domain]] $D$.
4 KB (614 words) - 06:23, 12 October 2023
$#A+1 = 127 n = 3
...cyclopedia/old_files/data/U095/U.0905430 Uniqueness properties of analytic functions
10 KB (1,496 words) - 08:27, 6 June 2020
$#A+1 = 17 n = 0
...= 17 : ~/encyclopedia/old_files/data/S090/S.0900500 Strip method (analytic functions)
4 KB (542 words) - 16:52, 19 February 2022
...clopediaofmath.org/legacyimages/o/o070/o070660/o0706604.png" /> the number of unknowns. The primary question for an overdetermined system is its solvabil
For example, an overdetermined system of linear algebraic equations
5 KB (715 words) - 08:35, 19 October 2014
$#A+1 = 22 n = 0
...1912 (cf. ). The sequence of Bernstein polynomials converges uniformly to a function $ f $
4 KB (588 words) - 10:58, 29 May 2020
$#A+1 = 30 n = 0
Consider a complex power series
4 KB (625 words) - 15:35, 4 June 2020
...C. T. Anger who in {{Cite|An}} studied the integral on the right hand side of \eqref{e:Anger} when the upper limit is $2\pi$ rather than $\pi$ . The Ange
...tion coincides with the Bessel function $J_\nu$ of order $n$ (cf. [[Bessel functions]]). For non-integer $\nu$ the following expansion is valid:
2 KB (361 words) - 09:02, 22 February 2014
* a criterion for the convergence of series of real numbers
...ers, used often to determine the convergence of power series at the radius of convergence
2 KB (382 words) - 12:44, 10 December 2013
''on conformal mapping of domains with variable boundaries''
One of the main results in the theory of conformal mapping of domains with variable boundaries; obtained by C. Carathéodory [[#References|[1]]].
4 KB (694 words) - 21:37, 22 December 2018
$#A+1 = 17 n = 1
...ential equation|Riemann differential equation]] as a result of the merging of two singular points. The point $ x = 0 $
4 KB (500 words) - 09:58, 26 March 2023
$#A+1 = 43 n = 0
...n to the problem of local inversion of holomorphic functions. In fact, let a function $ w = g(z) $
5 KB (731 words) - 06:29, 30 May 2020
$#A+1 = 72 n = 0
''singular point of multi-valued character''
5 KB (790 words) - 08:29, 20 January 2024
$#A+1 = 20 n = 0
$#C+1 = 20 : ~/encyclopedia/old_files/data/E036/E.0306850 Exhaustion of a domain,
2 KB (348 words) - 19:20, 17 January 2024
$#A+1 = 48 n = 0
...e of analytic functions subject to prescribed relations on the boundary of a domain.
4 KB (563 words) - 19:42, 5 June 2020
...th a cusp at the point $w=1$, touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. band is the Zhukovski
...n the $t$-plane to the exterior of $L'$. To obtain a Zhukovskii profile of a more general shape and disposition, the generalized Zhukovskii function is
4 KB (542 words) - 19:58, 4 January 2024
$#A+1 = 36 n = 0
of the extended complex plane $ \overline{\mathbf C} $,
5 KB (741 words) - 07:34, 10 April 2023
''of a domain $G$ in the $z$-plane''
...to a homeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.
1 KB (182 words) - 19:25, 13 December 2015
$#A+1 = 24 n = 0
...24 : ~/encyclopedia/old_files/data/I050/I.0500130 Ill\AAhposed problems in complex function theory
6 KB (900 words) - 22:11, 5 June 2020
$#A+1 = 47 n = 0
...mutative Banach algebra|Commutative Banach algebra]]). Much of the content of this work can be found in [[#References|[a3]]].
5 KB (677 words) - 13:07, 7 April 2023
$#A+1 = 13 n = 0
''in the theory of functions of a complex variable''
3 KB (354 words) - 04:11, 6 June 2020
$#A+1 = 73 n = 0
...al mapping that is induced by these functions and the initial coefficients of the power series that represent them.
8 KB (1,113 words) - 22:15, 5 June 2020
$#A+1 = 51 n = 0
Keldysh' theorem on approximating continuous functions by polynomials. Let $ f ( z) $
6 KB (831 words) - 22:14, 5 June 2020
A generalization of the
[[Zeta-function|zeta-function]] at the cost of introducing characters (cf.
2 KB (347 words) - 21:23, 9 January 2015
...le $x$ is defined in a neighborhood of a point $x_0$ and that there exists a finite or infinite limit
This limit is called the derivative of the function $f$ at the point $x_0$. If one sets $y=f(x)$,
4 KB (596 words) - 11:47, 5 July 2016
$#A+1 = 84 n = 0
...derivative|Dini derivative]]) of a given function of a complex variable at a given point. More precisely, let $ E $
6 KB (874 words) - 08:01, 6 June 2020
$#A+1 = 43 n = 0
A method in the geometric theory of functions of a complex variable that is used to solve extremal problems in function classes by representing
6 KB (949 words) - 06:44, 9 May 2022
A differential equation of the form
...d continuous function on the interval $-\infty<t<\infty$. A generalization of the Löwner equation is the Kufarev–Löwner equation:
3 KB (512 words) - 08:57, 29 August 2014
$#A+1 = 56 n = 0
of points of the complex plane $ \mathbf C $
7 KB (1,070 words) - 08:10, 6 June 2020
$#A+1 = 25 n = 1
$#C+1 = 25 : ~/encyclopedia/old_files/data/V096/V.0906150 Variation of a univalent function
4 KB (590 words) - 08:27, 6 June 2020
...it is possible to extract from $\mathcal F$ an at most countable sequence of pairwise disjoint sets $\{F_i\}$, $i=1,2,\dots$, such that
m_e\left[A\setminus\bigcup_{i=1}^{\infty}F_i\right]=0,
7 KB (1,055 words) - 05:43, 26 April 2024
$#A+1 = 101 n = 0
A rational function is a function $ w = R ( z) $,
8 KB (1,257 words) - 03:49, 4 March 2022
The ''pole of a function'' is
...ity $a=\infty$, the function $f(z)$ can be written as a [[Laurent series]] of special form:
3 KB (549 words) - 09:10, 18 January 2014
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/H048/H.0408380 Hypercomplex functions
5 KB (724 words) - 22:11, 5 June 2020
$#A+1 = 69 n = 0
...Napier number. This function is defined for any value of $ z $ (real or complex) by
6 KB (893 words) - 17:39, 20 January 2022
$#A+1 = 32 n = 0
$#C+1 = 32 : ~/encyclopedia/old_files/data/M063/M.0603990 Minimization of an area
2 KB (311 words) - 20:36, 22 December 2020
$#A+1 = 11 n = 0
A linear ordinary second-order differential equation in the complex domain
3 KB (443 words) - 18:25, 1 May 2024
$#A+1 = 108 n = 0
...principle]] for analytic functions to the case of functions that are given a priori as unbounded; it was first given in its simplest form by E. Phragmé
9 KB (1,334 words) - 08:06, 6 June 2020
$#A+1 = 77 n = 0
...eory)|Variational principles (in complex function theory)]]) in the theory of [[Conformal mapping|conformal mapping]], discovered by E. Lindelöf [[#Refe
6 KB (873 words) - 22:16, 5 June 2020
$#A+1 = 29 n = 0
A function $ f \in L _ {2} ( - \infty , + \infty ) $
4 KB (591 words) - 08:04, 6 June 2020
A fundamental theorem in complex analysis which states the following.
...athbb C$ a [[Holomorphic function|holomorphic funcion]], then the integral of $f(z)\, dz$ along any closed [[Rectifiable curve|rectifiable curve]] $\gamm
4 KB (659 words) - 13:04, 3 January 2014
$#A+1 = 28 n = 0
which is the so-called Gaussian probability integral. For a random variable $ X $
4 KB (598 words) - 19:23, 11 January 2024
$#A+1 = 32 n = 0
$#C+1 = 32 : ~/encyclopedia/old_files/data/I052/I.0502450 Inversion of a series
4 KB (554 words) - 20:40, 16 January 2024
...The majority of extremal properties of functions can be stated in the form of some exact inequality.
...theory (see [[#References|[6]]] and [[#References|[7]]]) and in the theory of numerical integration (see [[#References|[8]]]).
4 KB (575 words) - 20:24, 20 December 2016
$#A+1 = 42 n = 0
...nctions of a complex variable, formulated by B. Riemann and justified by H.A. Schwarz in the 19th century.
5 KB (635 words) - 06:45, 13 June 2022
$#A+1 = 12 n = 0
...neralizations (see [[#References|[1]]]). K. Löwner himself used the method of parametric representation on the class $ S $
3 KB (358 words) - 08:05, 6 June 2020
$#A+1 = 25 n = 0
A principle in the theory of [[Conformal mapping|conformal mapping]], proposed in 1928 by H. Grötzsch
4 KB (637 words) - 19:42, 5 June 2020
.../c/c026/c026400/c0264001.png" />. The applicability
of a number
of methods
of study is connected with the term "convexity" .
...ecting these points. Both definitions
of convexity are applied to the case
of vector spaces <img align="absmiddle" border="0" src="https://www.encycloped
15 KB (2,101 words) - 17:13, 7 February 2011
$#A+1 = 45 n = 1
be a bounded regular analytic function in the unit disc $ D = \{ {z } : {| z |
4 KB (555 words) - 19:29, 26 March 2023
A regular [[Univalent function|univalent function]]
...ng inequality expresses
a necessary and sufficient condition for convexity
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
19 KB (2,650 words) - 17:07, 7 February 2011
$#A+1 = 110 n = 0
A problem for the class of functions
9 KB (1,366 words) - 17:45, 4 June 2020
...encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634601.png" /> (or on
a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclop
...ww.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346010.png" /> is
a field with respect to the usual [[pointwise operation]]s followed by redefi
44 KB (5,974 words) - 22:47, 29 November 2014
$#A+1 = 59 n = 0
A solution of the [[Confluent hypergeometric equation|confluent hypergeometric equation]]
8 KB (976 words) - 17:46, 4 June 2020
$#A+1 = 70 n = 0
...unction]]). In other words, the problem of inverting the Abelian integrals of the first kind on the compact [[Riemann surface|Riemann surface]] $ F $
8 KB (1,183 words) - 22:14, 5 June 2020
$#A+1 = 39 n = 0
of the least upper bound of the modulus $ M( \alpha , \beta ) $
4 KB (618 words) - 13:00, 6 January 2024
$#A+1 = 125 n = 0
...al quantity, therefore it could be said that mathematical analysis studies functions and their generalizations by infinitesimal methods.
19 KB (2,844 words) - 08:57, 13 January 2024
An [[Isolated singular point|isolated singular point]] $z_0$ of an [[Analytic function|holomorphic function]] $f: U\setminus \{z_0\}\to \ma
...ed singular point $z_0$, any holomorphic function $f$ can be expanded into a Laurent series:
5 KB (819 words) - 10:11, 18 January 2014
An expression of the form
...the ''coefficients'' of the polynomial) and $k,l,\dots,t$ (the ''exponents of the powers'', which are non-negative integers) are constants. The individua
9 KB (1,497 words) - 10:44, 27 June 2015
$#A+1 = 142 n = 2
...edia/old_files/data/A013/A.0103000 Approximation of functions of a complex variable
18 KB (2,637 words) - 07:20, 26 March 2023
...i}$, can be represented in the unit disc $U=\{ z\in\C : \abs{z} < 1 \}$ by a Poisson–Stieltjes integral
where $\mu$ is a Borel measure concentrated on the unit circle $T=\{ z\in\C : \abs{z} = 1 \}
4 KB (706 words) - 19:19, 27 July 2012
$#A+1 = 22 n = 0
$#C+1 = 22 : ~/encyclopedia/old_files/data/M063/M.0603570 Method of boundary integration,
8 KB (1,130 words) - 08:00, 6 June 2020
''elliptic modular function, of one complex variable''
...://www.encyclopediaofmath.org/legacyimages/m/m064/m064430/m0644303.png" /> of the form
39 KB (5,287 words) - 17:07, 7 February 2011
$#A+1 = 32 n = 0
A numerical invariant $ \tau = \tau ( E) $
4 KB (516 words) - 06:48, 22 February 2022
$#A+1 = 11 n = 0
''of an analytic function $ f ( z) $''
1 KB (227 words) - 08:30, 20 January 2024
$#A+1 = 122 n = 0
...122 : ~/encyclopedia/old_files/data/B017/B.0107340 Boundary value problem, complex\AAhvariable methods
11 KB (1,622 words) - 06:29, 30 May 2020
...operator]], then so is $B$; if $A$ and $B$ are bounded operators, then $\|A\|=\|B\|$.
...also a sufficient condition for the unitary equivalence of operators with a pure point spectrum.
3 KB (374 words) - 11:04, 22 August 2014
$#A+1 = 126 n = 0
$#C+1 = 126 : ~/encyclopedia/old_files/data/A012/A.0102200 Analytic continuation
10 KB (1,475 words) - 18:21, 10 October 2023
$#A+1 = 24 n = 0
$#C+1 = 24 : ~/encyclopedia/old_files/data/A013/A.0103900 Attainable boundary point
2 KB (403 words) - 18:48, 5 April 2020
$#A+1 = 27 n = 0
...gorithm]]), in which the basic computing operation is the detection in $ A $
6 KB (915 words) - 08:14, 6 June 2020
$#A+1 = 130 n = 1
A function of a complex variable having a finite derivative. More precisely, a function $ f ( z) $
12 KB (1,760 words) - 08:01, 6 June 2020
$#A+1 = 49 n = 0
$#C+1 = 49 : ~/encyclopedia/old_files/data/A013/A.0103250 Argument, principle of the,
5 KB (749 words) - 18:48, 5 April 2020
$#A+1 = 68 n = 0
''Löwner's method of parametric representation of univalent functions, Löwner's parametric method''
7 KB (987 words) - 09:14, 6 January 2024
Out of 52 formulas, 51 were replaced by TEX code.-->
...ficients was posed (avoiding singular points, as needed even in the theory of ordinary differential equations). H. Lewy [[#References|[a5]]] found the fo
8 KB (1,236 words) - 19:09, 26 March 2023
$#A+1 = 85 n = 0
...lex plane $ \mathbf C $ (cf. [[Univalent function|Univalent function]]). A necessary and sufficient condition for $ f ( z) $
8 KB (1,121 words) - 08:59, 10 May 2022
$#A+1 = 68 n = 0
...[[power series]] in non-negative integral powers of the difference $ z - a $
8 KB (1,203 words) - 10:36, 20 January 2024
...re exists a path $L$: $z=z(t)$, $0\leq t<1$, $L\subset D$, terminating at $a$, i.e. so that
\[ \lim_{t → 1 - 0} z(t) = a \]
2 KB (383 words) - 13:44, 23 July 2015
$#A+1 = 32 n = 0
...[[Riemann sphere|Riemann sphere]] in the theory of functions of a complex variable.
2 KB (322 words) - 10:58, 29 May 2020
A principle expressing the connection between the lengths of curves belonging to some specific family and the area covered by this famil
...al length of the curves in $G$ on which $|f(z)|=\rho$; let $A$ be the area of $G$ and let
4 KB (608 words) - 19:06, 26 March 2023
$#A+1 = 69 n = 0
''of a meromorphic function $ w = f ( z) $
6 KB (869 words) - 22:17, 5 June 2020
''real-variable theory of, real-variable $\mathcal{H}^p$ theory''
...ern [[Harmonic analysis]]. They may be defined in terms of certain maximal functions.
6 KB (1,018 words) - 13:36, 26 December 2013
$#A+1 = 63 n = 0
...is a necessary and sufficient condition for quasi-analyticity in the sense of Hadamard, discovered by T. Carleman
7 KB (1,072 words) - 10:23, 2 June 2020
$#A+1 = 161 n = 0
$#C+1 = 161 : ~/encyclopedia/old_files/data/J054/J.0504050 Jacobi elliptic functions
18 KB (2,349 words) - 02:00, 20 June 2022
$#A+1 = 79 n = 0
...with at least three boundary points that is invariant under automorphisms of this domain.
7 KB (952 words) - 12:50, 13 January 2024
$#A+1 = 54 n = 0
...: ~/encyclopedia/old_files/data/V096/V.0906240 Variational principles (in complex function theory)
6 KB (783 words) - 18:43, 11 October 2023
Out of 32 formulas, 32 were replaced by TEX code.-->
Let $\sigma + i t$ be a complex variable. A finite sum
5 KB (706 words) - 17:03, 1 July 2020
$#A+1 = 111 n = 0
$#C+1 = 111 : ~/encyclopedia/old_files/data/F041/F.0401950 Function of bounded characteristic
8 KB (1,170 words) - 19:40, 5 June 2020
...tion in $D$ tend to a definite, finite or infinite, limit as $z$ tends to $a$ while remaining in $D$.
...in the annulus $D$ may be represented in a deleted neighbourhood of $a$ by a generalized Laurent series (Puiseux series):
2 KB (411 words) - 15:19, 29 December 2018
...ion for the finiteness of the velocity of the fluid at the rear sharp edge of the wing (see [[Zhukovskii function|Zhukovskii function]], Figure 2 and the
...D valign="top">[a2]</TD> <TD valign="top"> G. Birkhoff, "Hydrodynamics, a study in logic, fact and similitude" , Princeton Univ. Press (1960) pp. C
2 KB (289 words) - 19:58, 4 January 2024
$#A+1 = 39 n = 0
be a finitely-connected Jordan domain in the $ z $-
4 KB (585 words) - 19:15, 11 January 2024
$#A+1 = 127 n = 0
A process for obtaining a sequence of interpolation functions $ \{ f _ {n} ( z) \} $
12 KB (1,759 words) - 19:07, 11 January 2024
...a complex parameter with ${\rm Re}\, \nu > \frac{1}{2}$ and $x$ a complex variable.
The Struve functions of integral order $n$ is related to the [[Weber function|Weber function]] ${\b
5 KB (721 words) - 08:49, 22 February 2014
$#A+1 = 189 n = 0
A real-valued function $ u = u( z) $,
14 KB (2,065 words) - 08:06, 6 June 2020
where $a$ is a constant. Chebyshev's equation is a special case of the [[Hypergeometric equation|hypergeometric equation]].
...singular point]]) of the Chebyshev equation. Substituting the independent variable
1 KB (219 words) - 19:57, 21 November 2018
$#A+1 = 115 n = 0
...ia/old_files/data/C024/C.0204790 Conformal mapping, boundary properties of a
8 KB (1,180 words) - 17:46, 4 June 2020
...d of stating the conditions which ensure the existence (and non-existence) of such an extension over $k$.
...ns for the alternating groups; it was shown, in particular, that equations of the form
4 KB (586 words) - 04:07, 25 February 2022
$#A+1 = 20 n = 0
A property of a univalent conformal mapping $ f $
4 KB (564 words) - 21:51, 16 December 2020
$#A+1 = 52 n = 0
A method of using a sequence consisting of sequences
6 KB (881 words) - 20:32, 21 January 2021
An equation
of the form
...//www.encyclopediaofmath.org/legacyimages/l/l059/l059360/l05936010.png" />
of (1) defined for all <img align="absmiddle" border="0" src="https://www.ency
16 KB (2,227 words) - 17:26, 28 August 2018
$#A+1 = 103 n = 0
''of analytic functions in a domain''
7 KB (1,050 words) - 08:03, 6 June 2020
$#A+1 = 79 n = 0
$#C+1 = 79 : ~/encyclopedia/old_files/data/A012/A.0102800 Appell polynomials
11 KB (1,595 words) - 17:13, 2 January 2021
$#A+1 = 59 n = 0
''in a domain''
5 KB (677 words) - 08:27, 6 June 2020
''for measurability of a function of a real variable''
...$ there is a function $\phi$, continuous on $[a,b]$, such that the measure of the set
3 KB (436 words) - 19:43, 28 December 2017
$#A+1 = 32 n = 0
The following generalization of the [[Schwarz lemma|Schwarz lemma]]. Let $ w = f( z) $
4 KB (546 words) - 08:06, 6 June 2020
$#A+1 = 121 n = 1
A mapping of the complex space $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $
13 KB (1,875 words) - 13:58, 17 March 2023
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/E035/E.0305930 Equation of infinite order
5 KB (684 words) - 20:27, 10 January 2021
$#A+1 = 42 n = 0
The graph of the integral exponential function is illustrated in Fig..
6 KB (811 words) - 19:20, 11 January 2024
$#A+1 = 99 n = 0
$#C+1 = 99 : ~/encyclopedia/old_files/data/A010/A.0100220 Abelian function
11 KB (1,602 words) - 16:08, 1 April 2020
$#A+1 = 18 n = 0
Theorems which characterize the change in the argument under a [[Conformal mapping|conformal mapping]]. Rotation theorems in the class $
4 KB (570 words) - 19:31, 5 April 2023
''domain of type $C$, domain of type $S$''
...e harmonic function $\ln|\phi'(w)|$ can be written as the Poisson integral of its non-tangential boundary values $\ln|\phi'(e^{i\theta})|$:
3 KB (451 words) - 21:38, 15 December 2020
$#A+1 = 86 n = 0
...his method has been used for a long time in arithmetic to define sequences of numbers (progressions, Fibonacci numbers, etc.). Recursion plays an importa
13 KB (2,044 words) - 08:10, 6 June 2020
$#A+1 = 13 n = 0
$#C+1 = 13 : ~/encyclopedia/old_files/data/M063/M.0603700 Metric theory of functions
11 KB (1,633 words) - 08:00, 6 June 2020
$#A+1 = 71 n = 0
''Fourier–Stieltjes transform, of a probability measure $ \mu $''
8 KB (1,162 words) - 19:58, 19 January 2024
$#A+1 = 50 n = 0
$#C+1 = 50 : ~/encyclopedia/old_files/data/D033/D.0303920 Double of a Riemann surface
4 KB (635 words) - 19:36, 5 June 2020
$#A+1 = 54 n = 0
of functions
5 KB (790 words) - 11:06, 30 May 2020
$#A+1 = 33 n = 0
''indicator of an entire function''
4 KB (545 words) - 19:42, 5 June 2020
...and where it is stressed that only in this context the notion of a random variable becomes meaningful).
...$X$. This measure is uniquely determined by the [[distribution function]] of $X$:
6 KB (904 words) - 20:43, 1 January 2019
$#A+1 = 20 n = 0
The problem of extending a polynomial in $ z $
3 KB (480 words) - 06:30, 30 May 2020
$#A+1 = 54 n = 0
A series of functions used in the representation of automorphic forms and functions (cf. [[Automorphic form|Automorphic form]]; [[Automorphic function|Automorp
5 KB (778 words) - 08:25, 6 June 2020
$#A+1 = 64 n = 0
...f sets that are entirely contained in the range of values of each function of the corresponding class. Below some basic covering theorems are presented (
5 KB (681 words) - 17:31, 5 June 2020
$#A+1 = 83 n = 0
''of functions''
9 KB (1,339 words) - 19:49, 7 January 2024
$#A+1 = 120 n = 0
'' $ \theta $-function, of one complex variable''
14 KB (1,941 words) - 05:01, 23 February 2022
...he plane of the independent variable is ''regular singular'', if solutions of the equation loose analyticity, but exhibit at most polynomial growth at th
==Regularity as a growth condition for solutions==
5 KB (815 words) - 21:42, 31 July 2015
$#A+1 = 225 n = 0
A function $ u = u( x): D \rightarrow [ - \infty , \infty ) $
20 KB (2,972 words) - 12:24, 23 January 2021
$#A+1 = 209 n = 2
There are a number of theorems named after Karl Theodor Wilhelm Weierstrass (1815-1897).
22 KB (3,120 words) - 01:51, 24 February 2022
...mean the set of common zeros of a finite set of holomorphic functions) of complex dimension $n-1$.
...for some Borel measure $\mu$, that takes the value $+\infty$ at the points of $E$ and only at those points.
5 KB (817 words) - 17:25, 23 October 2017
A second-order linear ordinary differential equation
...ume complex values. The substitution $y=ux^{-1/2}$ yields the reduced form of equation \eqref{(1)}:
7 KB (1,130 words) - 20:00, 22 March 2024
$#A+1 = 51 n = 0
A method for solving functional equations of the type:
8 KB (1,085 words) - 08:29, 6 June 2020
$#A+1 = 183 n = 0
$#C+1 = 183 : ~/encyclopedia/old_files/data/A011/A.0101490 Algebraic function
20 KB (3,036 words) - 07:17, 15 June 2022
$#A+1 = 30 n = 0
A function $ u(x) = u(x _ {1} \dots x _ {n} ) $
5 KB (641 words) - 13:05, 23 February 2024
$#A+1 = 30 n = 0
The graphs of the functions $ y = \mathop{\rm ci} ( x) $
4 KB (588 words) - 19:18, 11 January 2024
$#A+1 = 56 n = 0
...: ~/encyclopedia/old_files/data/O070/O.0700350 Orthogonal polynomials on a complex domain
9 KB (1,315 words) - 20:15, 12 January 2024
$#A+1 = 64 n = 0
Picard's theorem on the behaviour of an analytic function $ f( z) $
8 KB (1,225 words) - 08:06, 6 June 2020
$#A+1 = 76 n = 0
== Privalov's theorem on conjugate functions ==
8 KB (1,054 words) - 08:07, 6 June 2020
...points close to $z_0$ but not coincident with it. The classical example of a movable singular point arises when considering the equation
...e surface $\{Q=0\}$ is irreducible and is projected along the $Ow$-axis on a region $\Omega\subset Oz$, then all points in the region $\Omega$ are movab
2 KB (271 words) - 14:56, 17 July 2014
$#A+1 = 45 n = 3
A regular analytic function of a complex variable $ z $,
7 KB (1,006 words) - 08:23, 26 March 2023
$#A+1 = 63 n = 0
''of one complex variable, elliptic modular form''
8 KB (1,125 words) - 07:04, 29 March 2024
$#A+1 = 34 n = 0
A homogeneous partial differential equation of the form
9 KB (1,300 words) - 19:02, 9 January 2024
$#A+1 = 37 n = 0
A parameter-dependent integral that gives a solution to the Schwarz problem on expressing an [[Analytic function|analyt
5 KB (698 words) - 18:20, 26 January 2022
$#A+1 = 24 n = 0
...la for rearranging the order of integration in iterated improper integrals of Cauchy principal value type (cf. [[Improper integral|Improper integral]]).
5 KB (699 words) - 08:06, 6 June 2020
$#A+1 = 39 n = 0
Originally, a measurable function was understood to be a function $ f ( x) $
3 KB (434 words) - 08:00, 6 June 2020
$#A+1 = 97 n = 0
A formula for the integral representation of holomorphic functions $ f ( z) $
8 KB (1,209 words) - 10:51, 20 January 2024
$#A+1 = 28 n = 1
of all functions $ f( z) $
3 KB (501 words) - 01:28, 26 April 2022
$#A+1 = 75 n = 1
...plane (except, possibly, at the point at infinity). It can be expanded in a power series
8 KB (1,251 words) - 20:13, 10 January 2021
$#A+1 = 31 n = 0
...ittlewood theorem in the theory of functions of a complex variable: If $ a _ {k} \geq 0 $,
4 KB (577 words) - 19:43, 5 June 2020
$#A+1 = 57 n = 0
''potential of a multi-pole''
6 KB (858 words) - 08:01, 6 June 2020
$#A+1 = 139 n = 0
...alytic continuations (cf. [[Analytic continuation|Analytic continuation]]) of an initial analytic function $ f = f( z) $
13 KB (1,972 words) - 20:02, 15 January 2024
...e $z$, $\operatorname{Si}(z)$, defined by \eqref{*}, is an entire function of $z$ in the $z$-plane.
For references, and the graph of the integral sine, see [[Integral cosine|Integral cosine]].
2 KB (327 words) - 21:27, 1 January 2019
$#A+1 = 161 n = 0
The main linear part of increment of a function.
15 KB (2,211 words) - 11:44, 26 March 2023
$#A+1 = 145 n = 0
...145 : ~/encyclopedia/old_files/data/I051/I.0501620 Integral representation of an analytic function
21 KB (2,949 words) - 22:12, 5 June 2020
...he elementary rule for the [[Derivative|derivative]] of the product of two functions. In its simplest form it states the following
...$I$ be an open interval and $x_0\in I$. Let $f,g : I \to \mathbb R$ be two functions which are [[Differentiable function|differentiable]] at $x_0$. Then $fg$ is
5 KB (757 words) - 10:34, 11 December 2013
$#A+1 = 206 n = 0
A regular or meromorphic function $ f $
24 KB (3,480 words) - 07:53, 14 January 2024
...eory of functions of a real variable typically involves a wide application of set-theoretic methods, along, naturally, with classical methods.
...h the formation of this theory took place in the 19th century (J. Fourier, A.L. Cauchy, N.I. Lobachevskii, P. Dirichlet, B. Riemann, P.L. Chebyshev, C.
11 KB (1,738 words) - 18:15, 24 March 2018
...t parallel are called ultra-parallel (divergent). So, for example, in Fig. a two lines are depicted passing through the point $z_1$ and both are paralle
...intersecting and divergent lines are defined as in the Poincaré model with a circular absolute. So, for example, in Fig. b two lines passing through the
5 KB (705 words) - 19:16, 14 August 2014
$#A+1 = 47 n = 0
...encyclopedia/old_files/data/F041/F.0401310 Fracture, mathematical problems of
10 KB (1,410 words) - 19:39, 5 June 2020
$#A+1 = 136 n = 0
$#C+1 = 136 : ~/encyclopedia/old_files/data/R081/R.0801560 Residue of an analytic function
16 KB (2,407 words) - 07:56, 11 January 2022
A map $f:X \to Y$ is called invertible if for every $y\in Y$ there exists one
...uthors still use the notation $f^{-1} (y)$), and it consists of the subset of $X$ defined by
10 KB (1,719 words) - 16:56, 30 November 2014
$#A+1 = 107 n = 1
A function of complex variables with the reproducing kernel property, defined for any domain $
10 KB (1,514 words) - 07:41, 26 March 2023
$#A+1 = 23 n = 0
...sums of a series in a domain that is larger than the domain of convergence of the series. The following theorems on over-convergence hold:
3 KB (438 words) - 12:39, 6 January 2024
$#A+1 = 182 n = 3
A function of a complex variable $ s = \sigma + it $
15 KB (2,181 words) - 11:50, 26 March 2023
A complex number is a number of the form $z=x+iy$, where $x$ and $y$ are real numbers (cf.
that is, a number whose square is equal to $-1$ (in engineering
14 KB (2,281 words) - 09:55, 26 March 2023
$#A+1 = 55 n = 0
...by Yu.V. Sokhotskii [[#References|[1]]], expressing the boundary values of a Cauchy-type integral. With more complete proofs but significantly later, th
9 KB (1,249 words) - 14:55, 7 June 2020
$\def\a{\alpha}$
An abbreviated form of notation in analysis, imitating the vector notation by single letters rathe
3 KB (512 words) - 06:16, 13 June 2022
...braic equations: Formulas expressing the solution of an arbitrary equation of degree $ n $
in terms of its coefficients using radicals do not exist for any $ n \geq 5 $.
6 KB (894 words) - 06:14, 26 March 2023
$#A+1 = 68 n = 0
...cyclopedia/old_files/data/U095/U.0905160 Undetermined coefficients, method of
9 KB (1,382 words) - 08:27, 6 June 2020
$#A+1 = 25 n = 0
be a function realizing a univalent [[Conformal mapping|conformal mapping]] of the disc $ \{ {z \in \mathbf C } : {| z | < 1 } \} $
3 KB (486 words) - 01:08, 19 March 2022
$#A+1 = 87 n = 0
...function inside a domain if certain bounds on the modulus on the boundary of the domain are known [[#References|[1]]], [[#References|[2]]]. Let $ D $
8 KB (1,149 words) - 19:43, 5 June 2020
$#A+1 = 64 n = 0
...odic function|double-periodic function]] that is meromorphic in the finite complex $ z $-plane. An elliptic function has the following basic properties.
9 KB (1,292 words) - 19:08, 20 January 2022
$#A+1 = 67 n = 0
of all fractional-linear transformations $ \gamma $
7 KB (1,031 words) - 18:33, 13 January 2024
==Liouville's theorem on bounded entire analytic functions==
of the complex variable $ z = ( z _ {1}, \dots, z _ {n} ) $
8 KB (1,240 words) - 04:55, 24 February 2022
...of transformations]] $ \Gamma $ of analytic transformations of a given complex manifold $ M $ :
...that are invariant under a discrete group $ \Gamma $ of automorphisms of this domain.
17 KB (2,502 words) - 06:16, 12 July 2022
$#A+1 = 78 n = 0
$#C+1 = 78 : ~/encyclopedia/old_files/data/A010/A.0100120 Abel\ANDGoncharov problem,
9 KB (1,324 words) - 11:49, 10 April 2023
$#A+1 = 92 n = 1
A solution of a [[Hypergeometric equation|hypergeometric equation]]
12 KB (1,576 words) - 01:14, 21 January 2022
$#A+1 = 247 n = 0
A Cauchy integral is a definite integral of a continuous function of one real variable. Let $ f (x) $
26 KB (3,804 words) - 08:11, 13 February 2022
Out of 101 formulas, 101 were replaced by TEX code.-->
...intimately reflect complex-analytic/geometric properties of the boundaries of such domains.
15 KB (2,167 words) - 16:10, 11 February 2024
$#A+1 = 200 n = 0
...cyclopedia/old_files/data/E035/E.0305250 Elasticity theory, planar problem of
21 KB (3,081 words) - 19:37, 5 June 2020
$#A+1 = 61 n = 0
...hematical logic — in the study of "infinite sets" , and in other branches of mathematics.
10 KB (1,463 words) - 22:12, 5 June 2020
If the indices $n_1,n_2,\ldots,$ of all non-zero coefficients of the power series
...has no [[Analytic continuation|analytic continuation]] across the boundary of this disc. Condition \ref{had} is known as Hadamard's condition; the gaps w
10 KB (1,520 words) - 12:44, 21 April 2012
...n $f$ is given on $X$ (and also that the variable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$.
...as a correspondence between two arbitrary sets (not necessarily consisting of numbers) was formulated by R. Dedekind in 1887 [[#References|[3]]].
34 KB (5,509 words) - 22:06, 28 January 2020
$#A+1 = 75 n = 0
The problem of determining in some region $ D $
11 KB (1,579 words) - 06:29, 30 May 2020
...sfinite diameter (see [[#References|[1]]]), somewhat later to the solution of the Carleman–Milloux problem (see [[#References|[2]]]) and were then used
...line passing through $w=w_0$, and if $r(E_f^*,w_0)$ is the interior radius of $E_f^*$ relative to the point $w_0$, then
5 KB (780 words) - 16:59, 14 February 2020
$#A+1 = 68 n = 1
...nciple|Lindelöf principle]], which employs the concept of subordination of functions. Let $ \omega ( x) $
5 KB (801 words) - 09:10, 6 January 2024
$#A+1 = 11 n = 1
''of a function $ f $
3 KB (417 words) - 20:00, 15 January 2024
$#A+1 = 291 n = 2
...encyclopedia/old_files/data/D031/D.0301930 Differential equation, partial, complex\AAhvariable methods
30 KB (4,345 words) - 09:02, 21 January 2024
$#A+1 = 263 n = 3
...encyclopedia/old_files/data/B017/B.0107330 Boundary properties of analytic functions
25 KB (3,676 words) - 08:39, 26 March 2023
$#A+1 = 254 n = 6
...ral generalization of that of a [[Univalent function|univalent function]]. A regular or meromorphic function $ f ( z) $
20 KB (3,020 words) - 17:24, 6 January 2024
$#A+1 = 56 n = 0
''boundary elements, prime ends, of a domain''
5 KB (847 words) - 22:16, 5 June 2020
$#A+1 = 69 n = 0
...pedia/old_files/data/P071/P.0701550 Parametric representation of univalent functions
8 KB (1,168 words) - 20:13, 12 January 2024
$#A+1 = 131 n = 0
A series of the form
11 KB (1,638 words) - 11:32, 16 April 2023
$#A+1 = 106 n = 2
...n class from incomplete information (for example, from its values on a set of points), using splines (cf. [[Spline|Spline]]).
13 KB (1,959 words) - 08:22, 6 June 2020
$#A+1 = 178 n = 1
In the wide sense it is a Laplace integral of the form
19 KB (2,911 words) - 18:54, 26 March 2023
$#A+1 = 72 n = 1
A hypothesis enunciated in 1916 by L. Bieberbach [[#References|[1]]]: For all functions $ f(z) $
10 KB (1,502 words) - 08:00, 26 March 2023
...pes of sequences of polynomials, or formal Laurent series, through the use of modern algebra techniques.
...ence of real numbers $a_0,a_1,a_2,\ldots$. Thus, if the sequences $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ satisfy
7 KB (1,142 words) - 20:34, 9 December 2015
Also generally called Bessel functions. Solutions $Z_\nu$ of the [[Bessel equation|Bessel differential equation]]
...eal or complex number. The cylinder functions are functions of one complex variable.
22 KB (3,786 words) - 06:49, 24 February 2024
...the general case the local uniformization theorem implies the existence of a finite resolving system (see [[#References|[3]]]).
...D valign="top">[3]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''3''' , Cambridge Univ. Press (1954) {{MR|0061846}}
3 KB (440 words) - 14:25, 1 May 2014
$#A+1 = 90 n = 0
A probability distribution which, for any $ n = 2, 3 \dots $
11 KB (1,489 words) - 09:00, 13 January 2024
$#A+1 = 18 n = 0
...rrently charged for an individual insurance contract be a weighted average of the form
5 KB (696 words) - 17:31, 5 June 2020
...ans just its syntax, and the possible semantics are called interpretations of the language.
...oncept of a formalized language lies at the bottom of Hilbert's concept of a [[Formal system|formal system]].
9 KB (1,407 words) - 19:01, 21 November 2018
$#A+1 = 19 n = 0
''on the surface of a liquid''
7 KB (1,020 words) - 20:50, 31 October 2023
Out of 173 formulas, 173 were replaced by TEX code.-->
...the classic paper [[#References|[a4]]] of G.-C. Rota and is a cornerstone of algebraic combinatorics (cf. also [[Combinatorics|Combinatorics]]).
11 KB (1,758 words) - 09:24, 10 November 2023
$#A+1 = 92 n = 0
A real stochastic process $ X = X( t) $,
8 KB (1,284 words) - 19:41, 5 June 2020
$#A+1 = 157 n = 0
...: ~/encyclopedia/old_files/data/A013/A.0103010 Approximation of functions of several real variables
21 KB (3,060 words) - 20:29, 10 January 2021
$#A+1 = 66 n = 1
...xpression in one of the equivalent forms $ a _ {0} + \sum _ {1} ^ {n} [ a _ {j} \cos ( jt ) + b _ {j} \sin ( jt ) ] $
9 KB (1,334 words) - 13:15, 26 March 2023
...a real-valued function $f$ of one variable is $n$ times differentiable at a point $x_0$, its Taylor formula has the form
...erentiable in some neighbourhood $(x_0-\delta,x_0+\delta)$, $\delta>0$, of a point $x_0$, then in this neighbourhood the remainder term can be written i
3 KB (536 words) - 09:04, 27 December 2013
Out of 87 formulas, 85 were replaced by TEX code.-->
''space of functions of bounded mean oscillation''
9 KB (1,406 words) - 19:56, 4 February 2024
$#A+1 = 308 n = 0
A continuous mapping preserving the form of infinitesimal figures.
28 KB (4,208 words) - 17:46, 4 June 2020
$#A+1 = 138 n = 0
...ction|residue of an analytic function]] of one complex variable to several complex variables. Let $ X $
11 KB (1,662 words) - 08:11, 6 June 2020
$#A+1 = 149 n = 2
$#C+1 = 149 : ~/encyclopedia/old_files/data/A012/A.0102430 Analytic space
22 KB (3,277 words) - 01:53, 19 January 2022
$#A+1 = 96 n = 0
.../encyclopedia/old_files/data/G043/G.0403760 Generalized almost\AAhperiodic functions
13 KB (1,809 words) - 19:41, 5 June 2020
$#A+1 = 71 n = 0
...ccessive integration with respect to different variables, i.e. an integral of the form
7 KB (1,043 words) - 01:21, 7 May 2022
...l operators is one of the few well-developed spectral theories for a class of non-self-adjoint operators. For the latter, see [[#References|[a3]]].
...r–Shaw inequality: If $T$ is a hyponormal operator (i.e. $[T^*,T]\geq 0$), of finite rational cyclicity $r(T)$, then:
6 KB (970 words) - 07:33, 18 March 2023
$#A+1 = 237 n = 0
A real-valued function $ u $,
21 KB (3,095 words) - 19:43, 5 June 2020
$#A+1 = 40 n = 0
$#C+1 = 40 : ~/encyclopedia/old_files/data/S087/S.0807230 Star of a function element,
5 KB (749 words) - 19:11, 19 June 2020
[[Category:Real functions]]
...ocess which yields a definition of integral of an unbounded function or of a function over an unbounded set, even when the function is not [[Absolutely
7 KB (1,076 words) - 12:30, 7 July 2013
Out of 11 formulas, 11 were replaced by TEX code.-->
All roots of a polynomial
5 KB (692 words) - 15:30, 1 July 2020
$#A+1 = 38 n = 0
''in a path-connected space''
4 KB (515 words) - 08:02, 6 June 2020
$#A+1 = 267 n = 0
$#C+1 = 267 : ~/encyclopedia/old_files/data/L059/L.0509490 Linear system of differential equations with periodic coefficients
22 KB (3,165 words) - 08:59, 21 January 2024
Out of 3 formulas, 2 were replaced by TEX code.-->
...Automatic control theory|Automatic control theory]]) is appraised in terms of the $ H ^ \infty $-
7 KB (1,101 words) - 16:08, 27 January 2024
...es $\Gamma(z) = (z-1)!$). It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product
...ed by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. [[Euler integrals]])
10 KB (1,513 words) - 09:50, 5 December 2012
$#A+1 = 179 n = 0
...ature. But the localization together with the accurate setting in the case of several variables was given by M. Sato, who employed [[Cohomology|cohomolog
17 KB (2,506 words) - 16:24, 22 February 2021
$#A+1 = 123 n = 0
''under conformal mapping of planar domains''
11 KB (1,524 words) - 02:42, 14 September 2022
...tric function]]), Sonin functions and many other functions — are solutions of linear differential equations with analytic coefficients.
Consider a system of $n$ equations in matrix notation:
16 KB (2,410 words) - 11:15, 28 January 2020
$#A+1 = 55 n = 0
$#C+1 = 55 : ~/encyclopedia/old_files/data/M064/M.0604490 Moduli of a Riemann surface
4 KB (668 words) - 06:30, 31 March 2023
...images/m/m062/m062160/m0621603.png" />-tuples
of non-negative real numbers
of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofm
...al order|partial order]] which occurs under various names in various parts
of mathematics: majority ordering, majorization ordering, specialization order
34 KB (4,631 words) - 18:28, 30 November 2016
$#A+1 = 79 n = 0
...-plane bending loading (thin plates bending). Limit equilibrium conditions of such bodies are also determined.
15 KB (2,200 words) - 14:56, 7 June 2020
...was designed. The language was also greatly influenced by the architecture of IBM computers and their operating systems.
...fying the requirements of particular applications. A PL/I program consists of independently compiled modules, called external procedures, which have an A
4 KB (681 words) - 12:48, 16 July 2014
...c space|analytic space]] related to the case where the ground field $K$ is a complete non-Archimedean normed field.
...peratorname{Max}B\to\operatorname{Max}A$, so that the affinoid spaces form a category.
6 KB (892 words) - 20:07, 22 December 2018
[[Orthogonal polynomials|Orthogonal polynomials]] defined in terms of generalized [[Hypergeometric series|hypergeometric series]] by
$${W_n(x^2;a,b,c,d)\over
5 KB (742 words) - 20:07, 24 December 2020
Out of 93 formulas, 93 were replaced by TEX code.-->
...ences|[a7]]] (see also [[#References|[a8]]]), and Wolff's boundary version of the [[Schwarz lemma|Schwarz lemma]] from 1926 [[#References|[a30]]].
14 KB (2,118 words) - 16:09, 11 February 2024
...ave an intuitive interpretation of them. Here are some (informal) examples of physical control system.
Nerve tissues consist of a structure of neurons and realize transformation of stimuli, coming from the outside and having effect on an organ.
12 KB (1,844 words) - 16:56, 15 April 2012
$#A+1 = 54 n = 0
...oundary properties of an analytic function in the unit disc and the metric of the Riemann surface onto which it maps the disc (see [[#References|[1]]], [
7 KB (1,075 words) - 19:29, 1 January 2021
Out of 61 formulas, 57 were replaced by TEX code.-->
...= \{ 0,1,2 , \dots \}$, the problem of finding a polynomial $p ( x , y )$ of degree $n$ which minimizes the $L^{2}$-norm
8 KB (1,192 words) - 08:49, 10 November 2023
$#A+1 = 72 n = 0
An analogue of the concept of a [[Characteristic function|characteristic function]]; it is used in the infi
7 KB (1,031 words) - 16:43, 4 June 2020
...the notion of a [[Limit|limit]] is defined. Here are the simplest examples of series.
==Simple series of numbers.==
29 KB (4,393 words) - 19:21, 27 January 2020
...udied in mathematics is continuously expanding; so this general definition of mathematics becomes ever richer in content.
...he first problems of mechanics and physics were already in this collection of fundamental mathematical ideas.
22 KB (3,256 words) - 21:54, 30 March 2012
$#A+1 = 77 n = 0
...development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differe
12 KB (1,635 words) - 14:54, 7 June 2020
$#A+1 = 307 n = 0
A function related to integral representations of solutions of boundary value problems for differential equations.
33 KB (4,788 words) - 08:53, 13 May 2022
$#A+1 = 321 n = 0
''of an analytic function $ w = f( z) $ of a complex variable $ z $''
34 KB (4,972 words) - 13:13, 6 January 2022
Out of 58 formulas, 58 were replaced by TEX code.-->
...functional equations. A main idea in it is to approximate the solution of a given problem by solving exactly an approximate problem.
20 KB (2,974 words) - 16:52, 1 July 2020
An ''elliptic curve'' is a non-singular complete
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
19 KB (3,251 words) - 20:37, 19 September 2017
$#A+1 = 85 n = 0
A numerical characteristic of a set of points in a Euclidean space $ \mathbf R ^ {n} $,
6 KB (914 words) - 10:37, 5 March 2022
$#A+1 = 347 n = 0
...s, and it is in fact such functions which are of interest, more than other functions, in applications.
37 KB (5,252 words) - 11:45, 26 March 2023
''For other meanings of the term 'Divisor' see the page'' [[Divisor (disambiguation)]]
...' is used as a generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer
16 KB (2,805 words) - 02:18, 6 January 2022
$#A+1 = 169 n = 0
...ms in potential theory are related to problems of the equilibrium shape of a rotating fluid and to problems in geophysics.
19 KB (2,714 words) - 13:11, 13 January 2024
$#A+1 = 35 n = 0
A problem involving partial differential equations in which the functions specifying the initial (boundary) conditions are not continuous.
7 KB (984 words) - 17:33, 5 June 2020
$#A+1 = 69 n = 1
A complete [[algebraic curve]] $ X _ {\widetilde \Gamma } $
8 KB (1,144 words) - 05:57, 13 February 2024
...tial]], [[integral]]. One of the simplest cases of a limit is the limit of a sequence.
==The limit of a sequence.==
32 KB (5,224 words) - 19:36, 25 March 2023
$#A+1 = 177 n = 0
...containing not only the unknown function itself, but also its derivatives of various orders.
33 KB (4,933 words) - 01:50, 23 January 2022
A partial sum of the [[Fourier series|Fourier series]] of a given [[Function|function]].
In the classical one-dimensional case where a function $f$ is integrable on the segment $[-\pi,\pi]$ and
6 KB (1,047 words) - 11:37, 13 February 2024
Out of 4 formulas, 4 were replaced by TEX code.-->
...ne special case, by F. Châtelet, that for an arbitrary Abelian variety $ A $
7 KB (1,109 words) - 16:59, 1 July 2020
...which certain real functions must be assigned "small" values if the values of the arguments are integers. Accordingly, Diophantine approximations are clo
...least one common integer argument may be studied at the same time. In such a case the Diophantine approximations are called simultaneous. Simultaneous D
54 KB (7,359 words) - 18:32, 31 March 2017
Classes
of analytic
functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...ory
of power series, linear operators, random processes, and in the theory
of extremal and approximation problems.
37 KB (5,073 words) - 18:20, 1 December 2014
An [[Algebraic variety|algebraic variety]] of dimension one. An algebraic curve is the most frequently studied object in
...ane affine algebraic curve. This is a set of points in an affine plane $ A _{k} ^{2} $
22 KB (3,307 words) - 17:02, 17 December 2019
The Wick products of random variables arise through an orthogonalization procedure.
is defined recursively as a polynomial in $f_1,\ldots,f_n$ of total degree $k_1+\cdots+k_n$ satisfying
13 KB (1,983 words) - 18:50, 11 December 2020
...areas are computed in a special metric corresponding to the peculiarities of the extremal problem under investigation. (About extremal problems in geome
...p method, H. Grötzsch obtained a number of classical results in the theory of conformal and quasi-conformal mappings (see, for example, [[Grötzsch theor
8 KB (1,208 words) - 17:22, 3 June 2016
...derived functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).
...a number of cases, to reduce the study of topological objects to the study of certain algebraic objects, as is done in analytic geometry (with the differ
12 KB (1,885 words) - 23:48, 23 April 2017
Out of 72 formulas, 72 were replaced by TEX code.-->
...ler measure $m ( P )$ is defined to be the average over the unit $n$-torus of $\operatorname { log } | P ( x _ { 1 } , \dots , x _ { n } ) |$, i.e.
7 KB (1,101 words) - 14:50, 27 January 2024
$#A+1 = 138 n = 0
...ble "perturbation theory" . The results obtained by using the two classes of methods complement each other.
21 KB (3,008 words) - 17:33, 5 June 2020
on a left $R$-module $X$'' is
a mapping $\def\phi{\varphi}\phi:X\times X \to R$ that is linear in the first
5 KB (831 words) - 17:13, 9 October 2016
...nd the complex analytic functions, which are commonly called ''holomorphic functions''. This entry concerns the latter: the reader is referred to [[Real analyti
...ons forms the main subject of the general theory of functions of a complex variable.
61 KB (9,850 words) - 19:04, 20 January 2022
$#A+1 = 51 n = 1
Consider a holomorphic [[Vector field|vector field]] with a singularity, i.e. of the form $ {\dot{x} } = Ax + \dots $
7 KB (1,085 words) - 08:13, 6 June 2020
$#A+1 = 146 n = 0
$#C+1 = 146 : ~/encyclopedia/old_files/data/A011/A.0101480 Algebraic equation
18 KB (2,778 words) - 16:09, 1 April 2020
$#A+1 = 329 n = 2
of a function $ f : G \rightarrow \Omega $,
25 KB (3,728 words) - 09:43, 26 March 2023
$#A+1 = 62 n = 0
...emannian manifold]] of constant negative curvature, or, more generally, of a discrete group $ G $
8 KB (1,145 words) - 22:11, 5 June 2020
Out of 266 formulas, 265 were replaced by TEX code.-->
...nd [[#References|[a8]]]; in particular, for the idea of points at infinity of $C$, see [[#References|[a1]]]. For an interplay between the geometry and th
16 KB (2,510 words) - 17:43, 1 July 2020
...lem of calculating the work performed by a force over an interval of time $a\leq t\leq b$ leads to this problem, as do other problems).
...lation of these interrelated forms of the integral constitutes the problem of [[Integral calculus|integral calculus]].
14 KB (2,170 words) - 17:21, 14 February 2020
$#A+1 = 145 n = 0
$#C+1 = 145 : ~/encyclopedia/old_files/data/A010/A.0100210 Abelian differential
11 KB (1,603 words) - 16:08, 1 April 2020
$#A+1 = 273 n = 0
An operator $ A $
25 KB (3,768 words) - 09:07, 14 June 2022
$#A+1 = 153 n = 0
...of some specific (function) or abstract Banach space, that is, an equation of the form
19 KB (2,803 words) - 19:40, 5 June 2020
Out of 66 formulas, 66 were replaced by TEX code.-->
...\xi _ { 1 } ^ { j_1 } \ldots \xi _ { n } ^ { j_n }$. To $P$ one associates a constant-coefficient linear partial differential operator $P ( D )$, obtain
8 KB (1,264 words) - 05:21, 19 March 2022
...rimes, studies the behaviour of number-theoretic functions, and the theory of algebraic and transcendental numbers.
==Distribution of prime numbers==
27 KB (4,516 words) - 18:38, 18 October 2014
Out of 99 formulas, 97 were replaced by TEX code.-->
...eratorname{SL} ( 2 , {\bf Z} )$ will be considered. If $f ( z )$ is a such a [[Modular form|modular form]], then
13 KB (1,907 words) - 07:36, 22 March 2023
...he collection of all first-order sentences true in a given structure forms a complete first-order theory $ T $.
For example, the models of the theory of the complex field $ ( \mathbf C ,\ +,\ \cdot ,\ 0,\ 1) $
11 KB (1,671 words) - 11:38, 22 December 2019
Out of 120 formulas, 119 were replaced by TEX code.-->
...lent definitions of the notion of vertex algebra and of the variant notion of vertex operator algebra are given below.
20 KB (2,919 words) - 00:57, 15 February 2024
$#A+1 = 75 n = 0
...else on a space dual to a space of this type. A differential expression is a mapping $ \lambda $
17 KB (2,519 words) - 19:35, 5 June 2020
$#A+1 = 239 n = 2
...n with the theory of nuclear spaces (cf. [[Nuclear space|Nuclear space]]), A. Grothendieck carried over the concept to locally convex spaces (see [[#Ref
24 KB (3,574 words) - 18:23, 21 January 2021
The branch of mathematics dealing with geometric objects connected with commutative rings
...he conversion of a tentative reasoning into a proof as for the formulation of such proofs in their most obvious and most general form.
29 KB (4,414 words) - 17:20, 17 December 2019
$#A+1 = 72 n = 0
...their gradient dynamical systems and the analogous list of critical points of codimension $ \leq 4 $(
11 KB (1,481 words) - 18:07, 1 June 2023
$#A+1 = 151 n = 0
To find a harmonic function $ u $
17 KB (2,472 words) - 08:26, 21 March 2022
...science of mathematics at a speed without parallel during the long history of mathematics.
...nce of discrete mathematics — and only the very beginning of the influence of discrete mathematics is witnessed now (1990).
13 KB (2,014 words) - 20:35, 31 December 2018
$#A+1 = 47 n = 0
...pplications of analysis, but also in theoretical concepts (e.g. the notion of compactness, etc.).
20 KB (2,978 words) - 19:15, 18 January 2024
$#A+1 = 90 n = 0
An equation containing finite differences of an unknown function. Let $ y ( n) = y _ {n} $
14 KB (1,916 words) - 19:59, 12 January 2024
$#A+1 = 267 n = 6
...gical structures encountered in practice. In a fairly broad interpretation of many-valued logic, logical calculi (cf. [[Logical calculus|Logical calculus
34 KB (5,105 words) - 19:20, 16 January 2024
$#A+1 = 30 n = 0
...nges in the intensity in radiation of the natural or induced radioactivity of rocks. The fields may be determined on the Earth's surface (surface methods
13 KB (1,884 words) - 19:41, 5 June 2020
...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut
...for Independent Data.''' For $i=1,\ldots,K,$ let $Y_i$ denote the response variable for the $i$th
15 KB (2,382 words) - 10:19, 10 March 2024
Galois cohomology studies the cohomology of a [[Galois group|Galois group]]. Let $M$ be an Abelian group, let $G(K/k)$ be
Galois group of an extension $K/k$ and suppose
17 KB (2,785 words) - 22:37, 23 December 2014
$#A+1 = 42 n = 0
...encyclopedia/old_files/data/E035/E.0305240 Elasticity, mathematical theory of
12 KB (1,716 words) - 19:37, 5 June 2020
$#A+1 = 386 n = 1
''of a set $ X $''
28 KB (4,564 words) - 07:37, 26 March 2023
$#A+1 = 35 n = 0
...o detect theoretically-predictable relations with a simultaneous choice of a concrete form for expressing them.
14 KB (2,040 words) - 19:36, 5 June 2020
$#A+1 = 120 n = 0
...d which is homeomorphic to a plane or a closed half-plane. It is the class of manifolds which are easiest to visualize; it includes the sphere, the disc,
20 KB (3,075 words) - 08:26, 6 June 2020
$#A+1 = 278 n = 0
...278 : ~/encyclopedia/old_files/data/S085/S.0805820 Small parameter, method of the,
35 KB (4,975 words) - 14:55, 7 June 2020
...eta-function one also distinguishes the generalized zeta-function $\zeta(s,a)$, the Dedekind zeta-function, the congruence zeta-function, etc.
...mplex $s$-plane for which $\sigma\geq1+\delta$, $\delta>0$. If $\sigma>1$, a valid representation is the [[Euler product|Euler product]]
45 KB (7,251 words) - 02:20, 29 June 2022
Out of 227 formulas, 227 were replaced by TEX code.-->
...rier hyperfunctions was proposed by M. Sato at the same time as his theory of hyperfunctions, in which the transformed objects can be interpreted natural
26 KB (3,903 words) - 17:01, 1 July 2020
$#A+1 = 145 n = 0
==Foundations of classical and quantum probability.==
22 KB (3,198 words) - 08:08, 6 June 2020
$#A+1 = 144 n = 0
...the elements, a rule for combining them and a means of using the structure of the control system to define the function which it realizes are given.
27 KB (4,142 words) - 14:55, 7 June 2020
A mapping which assigns to a point $ s $
of the base $ S $
12 KB (1,699 words) - 09:49, 20 December 2019
A ''linear algebraic group'' is
subgroup of a
14 KB (2,165 words) - 22:05, 5 March 2012
$#A+1 = 164 n = 0
...such as systems for deciding the equivalence of programs or the halting of a program.
20 KB (3,105 words) - 17:46, 4 June 2020
$#A+1 = 132 n = 0
Microlocal analysis considers (generalized, hyper-) functions, operators, etc. in the "microlocal" range. Here, "microlocal" means se
13 KB (1,805 words) - 19:06, 9 January 2024
...im of accelerating the calculations and of effectively using the resources of the computer.
...l programming a program generates a family of parallel-performed processes of information processing, which are completely independent or are related by
11 KB (1,714 words) - 03:17, 5 June 2016
$#A+1 = 144 n = 0
Iteration methods for the solution of non-linear equations.
20 KB (2,830 words) - 19:25, 9 January 2024
...{\pi}$, while the statement "the ratio of the length of the circumference of the circle to its diameter is greater than three and ten seventy-firsts and
...f mathematical notation was intimately bound up with the general evolution of mathematical concepts and methods.
18 KB (2,697 words) - 13:11, 13 December 2013
A [[Differential field|differential field]] $ F \supset F _{0} $
with a set $ \Delta $
13 KB (1,981 words) - 20:02, 17 December 2019
$#A+1 = 16 n = 0
...on symmetrical current flow around an equilateral wedge with an apex angle of $ 2 \alpha \pi $
8 KB (1,055 words) - 17:33, 5 June 2020
$#A+1 = 110 n = 0
The branch of geometry in which properties of figures are studied that are invariant under conformal transformations (cf.
14 KB (2,020 words) - 12:57, 13 January 2024
$#A+1 = 191 n = 1
A problem generated by the following equation, where $ x $
22 KB (3,082 words) - 19:17, 17 January 2024
$#A+1 = 124 n = 0
...lasses were introduced by L.S. Pontryagin [[#References|[1]]] in 1947. For a vector bundle $ \xi $
14 KB (1,987 words) - 19:15, 16 January 2024
$$A(x)\phi(x) + \int_DK(x,s)\phi(s)ds = d(x),\quad x\in D \label{1}$$
where $A$, $K$, $f$ are given functions, $A$ being called the coefficient, $K$ the kernel (cf.
14 KB (2,157 words) - 17:38, 3 September 2013
$#A+1 = 123 n = 0
...t case, a cybernetic system is reduced to a single element. An element $ A $
35 KB (5,424 words) - 11:17, 26 March 2023
$#A+1 = 152 n = 0
...itional semantics in the understanding of the disjunction $ A _ {0} \lor A _ {1} $(
12 KB (1,856 words) - 21:00, 4 June 2020
$#A+1 = 138 n = 0
...encyclopedia/old_files/data/P072/P.0702830 Plasticity, mathematical theory of
28 KB (4,176 words) - 05:30, 19 March 2022
...ematical interpretations, enabling one to use methods from different areas of mathematics.
...rties of the $ S $-matrix), or it can be considered from the point of view of $ C^{\ast} $-algebras and representation theory.
15 KB (2,298 words) - 22:02, 7 November 2017
$#A+1 = 144 n = 0
...(cf. [[Gas dynamics, numerical methods of|Gas dynamics, numerical methods of]]), which are quasi-linear.
24 KB (3,317 words) - 22:11, 5 June 2020
Out of 122 formulas, 122 were replaced by TEX code.-->
...er a phase space $\mathbf{R} ^ { 2 n }$, the Poisson bracket $\{ f , g \}$ of two smooth observables $f : \mathbf{R} ^ { 2 n } \rightarrow \mathbf{R}$ an
17 KB (2,575 words) - 16:58, 1 July 2020
$#A+1 = 39 n = 2
$#C+1 = 39 : ~/encyclopedia/old_files/data/O068/O.0608550 Optimization of computational algorithms
12 KB (1,798 words) - 20:07, 12 January 2024
addition. Moreover, an associative algebra should be a vector space
over a fixed field $F$, and the multiplication in it is compatible
11 KB (1,726 words) - 20:09, 15 December 2020
$#A+1 = 215 n = 1
...lian extensions (finite Galois extensions having Abelian Galois groups) of a field $ K $
17 KB (2,620 words) - 07:48, 13 February 2024
$#A+1 = 165 n = 2
...values at each point [[#References|[a7]]] — is that of an infinite system of coordinates on which to base an infinite-dimensional calculus. More precise
27 KB (3,916 words) - 19:20, 13 January 2024
$#A+1 = 211 n = 2
...multiplicative [[Cobordism|cobordism]] invariants taking values in a ring of modular forms. The following is an attempt to present the simplest case —
16 KB (2,293 words) - 19:37, 5 June 2020
$#A+1 = 144 n = 0
...pace under the action of some random mechanism. The state space is usually a $ d $-
18 KB (2,649 words) - 20:12, 12 January 2024
...fully undetermined or the text merely serves as material for the synthesis of an algorithm.
...occurrence of such a lexeme is said to be defining; the other occurrences of the lexeme in its scope are called applied.
12 KB (1,804 words) - 16:56, 7 February 2011
$#A+1 = 21 n = 1
...nly in the context of the application to the conditions for the minimum of a functional.)
14 KB (2,052 words) - 08:27, 6 June 2020
A singular point of an analytic function $ f(z) $
...tacle to the [[Analytic continuation|analytic continuation]] of an element of the function $ f(z) $
66 KB (9,825 words) - 01:45, 23 June 2022
$#A+1 = 90 n = 1
Divisors of the form
13 KB (1,821 words) - 09:04, 10 April 2023
$#A+1 = 219 n = 0
...art of modern mathematical analysis in which the basic purpose is to study functions $ y = f ( x) $
36 KB (5,132 words) - 08:10, 30 January 2022
$#A+1 = 128 n = 1
$#C+1 = 128 : ~/encyclopedia/old_files/data/A014/A.0104190 Autonomous system
13 KB (1,960 words) - 07:35, 26 March 2023
$#A+1 = 64 n = 0
...mply-connected complete Riemannian manifold of constant curvature, i.e. of a Euclidean space $ E ^ {n} $,
12 KB (1,735 words) - 19:59, 15 March 2023
$#A+1 = 112 n = 0
...encyclopedia/old_files/data/S087/S.0807380 Statistical experiments, method of
13 KB (1,879 words) - 11:25, 31 January 2022
$#A+1 = 222 n = 0
...obability]]) of certain random events are used to deduce the probabilities of other random events which are connected with the former events in some mann
42 KB (6,282 words) - 08:07, 6 June 2020
Out of 284 formulas, 257 were replaced by TEX code.-->
...yclopediaofmath.org/legacyimages/w/w120/w120110/w1201103.png"/> defined on functions $u ( x )$ as
30 KB (4,520 words) - 19:54, 6 February 2024
...number of degrees of freedom. Quantum field theory unifies the description of fields and particles, which, in classical physics, appear as two distinct e
...the example of an electromagnetic field, as this is the only field having a clear content, both in the classical and the quantum case.
35 KB (5,270 words) - 23:26, 6 December 2016
$#A+1 = 373 n = 5
$#C+1 = 373 : ~/encyclopedia/old_files/data/D033/D.0303530 Distribution of prime numbers
36 KB (5,530 words) - 08:29, 14 January 2024
$#A+1 = 122 n = 0
...or with an invertible principal symbol (see [[Symbol of an operator|Symbol of an operator]]).
12 KB (1,764 words) - 05:06, 24 February 2022
$#A+1 = 194 n = 0
A [[stochastic process]] $ X( t) $
24 KB (3,614 words) - 08:52, 21 January 2024
A matrix is a rectangular array
consisting of $m$ rows and $n$ columns, the
18 KB (3,377 words) - 17:54, 2 November 2013
...iscussed but ten of them (problems 1, 2, 6, 7, 8, 13, 16, 19, 21, 22). For a translation, see [[#References|[a19]]]. These 23 problems, together with sh
Cantor's problem on the [[Continuum, cardinality of the|cardinal number of the continuum]].
29 KB (4,109 words) - 19:54, 18 March 2018
$#A+1 = 133 n = 2
An equation of the form
30 KB (4,331 words) - 16:42, 20 January 2022
$#A+1 = 88 n = 2
...ielded elegant and conclusive results, but resulted in a mutual enrichment of the two topics [[#References|[3]]].
24 KB (3,602 words) - 11:48, 26 March 2023
Out of 145 formulas, 144 were replaced by TEX code.-->
...ruct elements $c _ { L } \in H ^ { 1 } ( G ( \overline { K } / K ( L ) ) ; A )$ such that:
19 KB (2,901 words) - 17:41, 25 November 2023
...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut
<ref> Department of Statistics and Genetics Institute,
19 KB (2,851 words) - 07:04, 7 March 2024
$#A+1 = 144 n = 7
...|conformal mapping]]. A numerical characterization of the distortion under a mapping $ f : D \rightarrow D ^ \prime $
28 KB (3,937 words) - 14:54, 7 June 2020
$#A+1 = 154 n = 4
A general term referring to the inequality $ 4 \pi V \leq F ^ { 2 } $
22 KB (3,071 words) - 22:13, 5 June 2020
$#A+1 = 37 n = 0
...onstructive trend in mathematics) is often applied in a more general sense of the word, referring to this Soviet approach to constructive mathematics. In
20 KB (2,913 words) - 07:40, 14 August 2023
...a system of functions $\{\phi_n\}$ which are orthonormal on an interval $(a,b)$.''
...ent can be relaxed by replacing it by another which ensures the existence of all the integrals in \ref{eq1}.
21 KB (3,473 words) - 19:50, 26 April 2012
$#A+1 = 202 n = 0
is a unit vector, $ \phi = \phi ( x, \Omega , v) $
17 KB (2,415 words) - 09:04, 13 January 2024
Out of 184 formulas, 184 were replaced by TEX code.-->
...tivating example (see [[#References|[a13]]], in which regularity estimates of the heat kernel are deduced).
21 KB (3,112 words) - 08:01, 6 February 2024
Out of 88 formulas, 86 were replaced by TEX code.-->
...storically, it is usually connected with the paper [[#References|[a1]]] by A.P. Calderón (1964), but its basic idea was known before.
10 KB (1,553 words) - 17:45, 1 July 2020
...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut
population or a part of it, such as the residents of a country or the
20 KB (2,970 words) - 20:18, 12 March 2016
$#A+1 = 85 n = 0
...s well as the development of efficient numerical algorithms. In tomography a fundamental role is played by the [[Radon transform|Radon transform]] on Eu
16 KB (2,342 words) - 17:25, 13 June 2020
...hich the quotient space $\Gamma\setminus G$ has finite volume in the sense of the measure induced by the left-invariant
...ure|Haar measure]] on the group $G$. The concept of lattices includes that of uniform discrete subgroups, for which the quotient space $\Gamma\setminus G
20 KB (3,146 words) - 02:52, 23 July 2018
$#A+1 = 147 n = 4
$#C+1 = 147 : ~/encyclopedia/old_files/data/A110/A.1100380 Adequacy theorem
17 KB (2,488 words) - 06:37, 26 March 2023
Out of 154 formulas, 152 were replaced by TEX code.-->
...r enumeration questions about the (mostly non-arithmetical) concrete types of mathematical objects just alluded to.
24 KB (3,738 words) - 07:41, 7 February 2024
''measure of a set''
...a real variable in connection with the study and improvement of the notion of an [[Integral|integral]].
46 KB (7,065 words) - 19:30, 1 January 2021
...pect to the equivalence relation. The ''normal form'' of an object $M$ is a "selected representative" from the class $[M]$, usually possessing some ni
...orm]] and [[Conjunctive normal form|conjunctive normal form]] for Boolean functions.
30 KB (4,754 words) - 19:19, 14 November 2023
[[Cohomology]] of
[[Coherent sheaf|coherent sheaves]]. Let $X$ be a non-singular
64 KB (9,418 words) - 12:44, 8 February 2020
Out of 147 formulas, 143 were replaced by TEX code.-->
...$\mathcal{H}$, the upper half-plane. A fundamental region, $\mathcal{R}$, of $\Gamma$ is required to have finite hyperbolic area; this is equivalent to
13 KB (1,993 words) - 07:12, 15 February 2024
$#A+1 = 394 n = 0
...encyclopedia/old_files/data/L059/L.0509160 Linear differential equation in a Banach space
32 KB (4,803 words) - 17:49, 6 January 2024
...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut
'''Summary''' Poisson, the apostle of Laplacian science, was the master of
11 KB (1,710 words) - 19:08, 6 March 2024
$#A+1 = 263 n = 0
...each point. These concepts of curvature are connected with the examination of deviations which are small to the second order; hence the object in questio
26 KB (4,098 words) - 17:31, 5 June 2020
A Drinfel'd $A$-module, (where $A$ is an appropriate
[[Ring|ring]]) over a
19 KB (3,204 words) - 20:11, 14 April 2012
...m. The local classification problem is to describe the equivalence classes of local dynamical systems, providing, if possible, the simplest or most conve
...e field (map). This hope is partly justified, see [[Algebraic decidability of local classification problems]].
37 KB (5,881 words) - 19:10, 24 November 2023
$#A+1 = 498 n = 0
The normal form of a matrix $ A $
51 KB (7,267 words) - 07:39, 14 January 2024
...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut
of random variation. The widest range of applications is found for them
23 KB (3,576 words) - 22:10, 12 March 2016
...'the discretization program'".</ref>These efforts took place over a period of about 50 years, with the following results:
# the establishment of fundamental concepts related to limits
91 KB (14,186 words) - 16:58, 23 November 2023
$#A+1 = 178 n = 8
...: ~/encyclopedia/old_files/data/N066/N.0606200 Negative curvature, surface of
48 KB (7,189 words) - 08:02, 6 June 2020
...pyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribut
Biostatistics Unit, Department of Public Heath
17 KB (2,653 words) - 16:21, 10 March 2024