...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