...of]]; [[Functions of a complex variable, theory of|Functions of a complex variable, theory of]]).
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''simple function (of a complex variable)''
...older, names for a [[Univalent function|univalent function]] of a complex variable.
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''bound occurrence of a variable''
...nt of a bound variable, where it occurs, by a new (for a given expression) variable leads to an expression with the same meaning. For example, in the expressio
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...ign to a [[Convex function (of a real variable)|convex function (of a real variable)]].
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...ual variable of the corresponding type (here also it is necessary to avoid variable collisions).
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$#C+1 = 15 : ~/encyclopedia/old_files/data/P074/P.0704380 Predicate variable,
''second-order variable''
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The curve representing the ends of the variable vector $x(t)$ ($t$ is a real variable, such as time) whose origin for all $t$ is a given fixed point $O$.
...ograph. For example, if the velocity of a moving point is represented by a variable vector $\mathbf{v}$, then by drawing the values of $\mathbf{v}$ at differen
1 KB (179 words) - 19:16, 12 April 2017
...ude characterizing the [[Dispersion|dispersion]] of the values of a random variable $ X $.
is the distribution function of the random variable. Another variable which is also occasionally considered is the Gini dispersion coefficient
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''surplus variable''
A non-negative variable $y_i$ that is introduced for a (linear) constraint $\sum_ja_{ij}x_j\leq b_i
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$#C+1 = 36 : ~/encyclopedia/old_files/data/F041/F.0401660 Free variable,
''free occurrence of a variable''
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...and the set of conditional probabilities $\mathrm P_\theta$ of the random variable $X$ given $\Theta=\theta$. According to the [[Bayes formula|Bayes formula]]
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A function $f(z)$ of a complex variable $z$ which is single-valued in this domain and which has a finite derivative
[[Category:Functions of a complex variable]]
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.../> is a [[Convex function (of a real
variable)|convex function (of a real
variable)]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmat
3 KB (355 words) - 17:19, 7 February 2011
A [[periodic function]] $f(z)$ of the complex variable $z$ all periods $p$ of which are integer multiples of a single unique funda
...riod $2\omega\in X$. A non-constant continuous periodic function of a real variable is necessarily simply-periodic.
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The argument of a function is the variable (also called the independent variable) on which the value of the function depends.
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...ibly others) in accordance with the following rules: 1) each propositional variable is a propositional formula; and 2) if $A,B$ are propositional formulas, the
1 KB (139 words) - 13:57, 30 December 2018
...esis is called the hypothesis distribution. E.g., if one observes a random variable $X$, then the statement "X is subject to the Poisson law with parameter 1"
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...mber for each set of values of the parameters ($x$ in this term is a bound variable). Syntactically, terms are characterized by the fact that they can be subst
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...on a [[probability space]] $(\Omega,\mathcal{F},\mathbb{P})$, to a random variable $X$, defined in the following way: $X_n \stackrel{\mathrm{P}}{\rightarrow}
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''in the theory of functions of a complex variable''
[[Category:Functions of a complex variable]]
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A function that does not change sign when the sign of the independent variable is changed, i.e. satisfying the condition $f(-x)=f(x)$. The graph of an eve
A function that does change sign when the sign of the independent variable is changed, i.e. satisfying the condition $f(x)=-f(-x)$, is called an [[Odd
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...s also defined for discrete distributions: If the values $x_k$ of a random variable $X$ with distribution $p_k = \mathsf{P}(X = x_k)$ are arranged in increasin
...istics)]] the mode acts as a measure of location of the values of a random variable. For distributions which are unimodal and symmetric with respect to some po
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Here $z$ is a complex variable, $m$ is an integer and $n$ is a natural number. The series converge in the
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)</TD><
1 KB (199 words) - 17:44, 1 November 2014
...f. [[Convex function (of a complex
variable)|Convex function (of a complex
variable)]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex
variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
4 KB (623 words) - 17:10, 7 February 2011
...em is one of the classical results in the theory of functions of a complex variable of the type of [[Distortion theorems|distortion theorems]].
...TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD
3 KB (454 words) - 08:12, 6 June 2020
The inverse hyperbolic functions of a real variable $ x $
The inverse hyperbolic functions of a complex variable $ z $
3 KB (487 words) - 19:15, 17 January 2024
...i=1,\dotsc,m$; $j=1,\dotsc,m_i$) is either a variable or the negation of a variable, and $\bot$ is the logical symbol denoting falsehood. For each propositiona
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...n of values of the [[Normal distribution|normal distribution]] of a random variable at a distance from its [[Mathematical expectation|mathematical expectation]
Let $X$ be a normally $N(a,\sigma^2)$ distributed random variable. For any $k>0$,
1 KB (223 words) - 00:44, 24 December 2018
...ing time (relative to this family of subalgebras) is a mapping (a [[random variable]]) $\tau : \Omega \rightarrow T \cup \{\infty\}$ such that
...iable. This condition has the interpretation that the (time-valued) random variable $\tau$ has no knowledge of the future, since the $\sigma$-algebra $\mathcal
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The probability distribution of the random variable
is a random variable subject to the standard normal law $ N( 0, 1) $
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is a real or complex variable. If $ a _ {0} \neq 0 $,
...lane, that is, it is an [[Entire function|entire function]] of the complex variable $ z $,
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...mula is a formula; and b) if $A$ and $B$ are formulas and $y$ is an object variable, then $(\neg A)$, $(A\mathbin\&B)$, $(A\lor B)$, $(A\supset B)$, $(\forall
2 KB (252 words) - 16:42, 30 December 2018
...f Cauchy distributions is closed under linear transformations: If a random variable $ X $
then the random variable $ Y = aX + b $
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cf. [[Random variable|Random variable]]). For $ m < N $,
4 KB (507 words) - 22:10, 5 June 2020
...makes it possible to reduce the study of the deformation of surfaces with variable $ K > 0 $
...r-reaching analogy between the properties of deformations of surfaces with variable and constant positive Gaussian curvature.
1 KB (168 words) - 08:28, 6 June 2020
be a random variable having a continuous and strictly increasing distribution function $ F $.
Then the random variable $ Y = F ( X) $
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The distribution of a discrete random variable assuming non-negative integral values $ m = 0, 1 \dots $
The random variable equal to the number of independent trials prior to the first successful out
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''conditional expectation, of a random variable''
...ctation of $X$ with respect to $\mathfrak{B}$ is understood to be a random variable $\mathsf{E}(X\, |\, \mathfrak{B})$, measurable with respect to $\mathfrak{B
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...depending on a continuous or discrete argument $\tau$, relative to another variable $\phi(\tau)$ whose behaviour, as a rule, is assumed to be known. In general
...lign="top"> A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)</TD></TR><TR><TD valign="top
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A property of the dependence of one random variable on another, defined by the fact that the conditional variance is equal to a
280 bytes (45 words) - 17:14, 7 February 2011
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
[[Category:Functions of a complex variable]]
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a random variable defined on $ ( \Omega , {\mathcal A} ) $
is called a (regular) conditional distribution of the random variable $ X $
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...$F(x) = \mathbf{P}\{X \le x\}$ be the distribution function of the random variable $X$. Then the probability distribution for the range is
1 KB (196 words) - 12:46, 14 February 2020
...im_i}$ do not contain both the formulas $p$ and $\neg p$, where $p$ is any variable. For any propositional formula $A$ it is possible to construct an equivale
2 KB (307 words) - 16:22, 18 November 2023
with respect to the variable $ x _ {1} $
in the single variable $ x _ {1} $,
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be a continuously-distributed random variable for which the probability density $ p( x) $,
...tuation, when instead of the realizations of the initial continuous random variable $ X $,
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A function $f$ that changes sign when the independent variable changes sign, i.e. $f(-x)=-f(x)$ for all values in the domain of $f$. The g
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\Var X$ of the deviation of a [[Random_variable | random variable]] $X$ from its [[Mathematical expectation|mathematical expectation]] $\Ex X
...us functions of a random variable $X$ in terms of the distribution of this variable on the set of real numbers are of importance (cf. [[Mathematical expectatio
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...TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD
...top">[a1]</TD> <TD valign="top"> J.B. Conway, "Functions of one complex variable" , Springer (1978)</TD></TR></table>
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...culation of an area or an [[Integral|integral]] (of a function of a single variable).
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''on conformal mapping of domains with variable boundaries''
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) (Translated from Russian)</TD></TR></table>
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...pplied to predicates not with respect to the whole range of a given object variable, but with respect to a part of it defined by a [[Predicate|predicate]] $R(x
that is, the predicate $P(x)$ is true for all values of the variable $x$ satisfying the predicate $R(x)$. The proposition $(\exists x)_R(X)P(x)$
1 KB (212 words) - 07:30, 23 August 2014
...lytic functions (cf. [[Analytic function|Analytic function]]) of a complex variable $z$ in a domain $D$, let a simple closed piecewise-smooth curve $\Gamma$ to
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><TD va
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...he membership of a distribution function (probability density) of a random variable in a certain set of distribution functions containing more than one element
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...contact of $q$ and $l$ is said to be equal to $n$ if the segment $QL$ is a variable of $(n+1)$-st order of smallness with respect to $MK$ (see Fig., where $QL$
The phrase "QL is a variable of the variable of n+1-st order of smallness with respect to another variablen+1-st order o
2 KB (418 words) - 13:16, 29 April 2014
...n to the problem of local inversion of a holomorphic function of a complex variable. The original solution of the inversion problem, given by J.L. Lagrange in
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...o functions of several variables of the [[Variation of a function]] of one variable, proposed by
($TV (f\circ \gamma)$ is then the classical total variation of the real variable function $f\circ \gamma$, see [[Variation of a function]]).
2 KB (323 words) - 09:38, 16 August 2013
...ariable. It consists in transforming the integral by transition to another variable of integration. For the [[Definite integral|definite integral]] the formula
...a \eqref{e:change_of_var} can be generalized to integrals in more than one variable: we refer to [[Change of variables in an integral]] and [[Area formula]].
2 KB (308 words) - 14:34, 17 November 2012
is any variable not appearing as a free variable in $ \phi ( x) $
the variable $ y $
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In probability theory, the generating function of a [[Random variable|random variable]] $ \xi $
The generating function of a random variable $ \xi $
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...an even [[Convex function (of a real variable)|convex function (of a real variable)]] which is increasing for $u$ positive, and
2 KB (377 words) - 14:21, 11 August 2014
...robability distribution]]. For a continuously-distributed symmetric random variable $ X $
of the random variable around its median, [[#References|[a1]]].
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...a [[Probability distribution|probability distribution]] (e.g. of a random variable), the corresponding [[Distribution function|distribution function]] or the
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...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><T
...top">[a1]</TD> <TD valign="top"> J.B. Conway, "Functions of one complex variable" , Springer (1978)</TD></TR></table>
2 KB (233 words) - 19:38, 5 June 2020
...of dependence between random variables. The correlation ratio of a random variable $ Y $
relative to a random variable $ X $
2 KB (236 words) - 17:31, 5 June 2020
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
[[Category:Functions of a complex variable]]
2 KB (264 words) - 21:15, 1 November 2014
...identity for [[Holomorphic function|holomorphic functions]] of one complex variable. In the case of the unit circle one has the following relationship between
...n="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a complex variable" , '''1–3''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}}
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...e degree [[field extension]] of the field of [[rational function]]s in one variable over a [[finite field]] of constants or a finite extension of the field $\m
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Substitutions of the variable $x=x(t)$ in an integral
...$x$ and $\sqrt{ax^2+bx+c}$ to be expressed rationally in terms of the new variable $t$.
2 KB (366 words) - 06:13, 10 April 2023
of one complex variable $ z $,
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)</TD><
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...mbers themselves. This is common practice, in particular, when considering variable quantities. If some given concrete quantity, say the length $ l $
remains constant). This time-variable number $ x $
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...of functions of a complex variable|approximation of functions of a complex variable]]; it was obtained by P. Montel .
...<TD valign="top"> P. Montel, "Leçons sur les séries de polynomes à une variable complexe" , Gauthier-Villars (1910)</TD></TR><TR><TD valign="top">[2]</TD>
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of a complex variable $ z $
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapts. 2; 8 (Translated from Russian)</
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periodic in each variable, that are either continuous or $ p $-
in the variable $ x _ {i} $(
4 KB (588 words) - 10:58, 29 May 2020
Dependence of the mean value of some random variable on another variable or on several variables. If, for example, for every value $ x = x _ {i} $
of a random variable $ Y $,
10 KB (1,524 words) - 08:10, 6 June 2020
...as a number $m$ such that $F(m)\leq1/2$ and $F(m+0)\geq1/2$. Every random variable has at least one median. If $F(x)=1/2$ for all $x$ in a closed interval, th
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then the random variable $ T ^ {2} = Y ^ { \prime } S ^ {-} 1 Y $
where the random variable $ t _ {n} $
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...or the relevant property) is called non-predicative if it contains a bound variable such that the object to be defined falls within its range of applicability.
...able ranging over all sets. In this formula $R$ is a possible value of the variable $x$. In mathematics non-predicative definitions are widely used. For exampl
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A dimensionless measure of the spread of the distribution of a random variable. This coefficient may be defined in several ways. In practice, its most fre
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...of functions of a complex variable|Approximation of functions of a complex variable]]).
Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis. A theorem analogous to Runge's theorem is valid
3 KB (531 words) - 18:58, 1 May 2014
...ial. An example of a serial ring is the ring of formal power series in one variable over a field.
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...th.org/legacyimages/o/o070/o070660/o0706609.png" /> is a polynomial in one
variable (an ordinary differential equation) or several variables (a partial differe
If (1) is a system of partial differential equations with
variable coefficients <img align="absmiddle" border="0" src="https://www.encyclopedi
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...opediaofmath.org/legacyimages/c/c026/c026020/c0260202.png" />, to a random
variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
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...TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD
...D> <TD valign="top"> C. Carathéodory, "Theory of functions of a complex variable" , '''1''' , Chelsea, reprint (1983) pp. 170 (Translated from German)</T
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''mean value, of a random variable''
...ble. In the most general setting, the mathematical expectation of a random variable $ X( \omega ) $,
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[[Probability distribution|probability distribution]] of a random variable $X$ which takes non-negative integer values, defined by the formula
...sample — has a negative hypergeometric distribution \eqref{*}. The random variable <math>X+m</math> — the size of the sample — also has a negative hyperge
3 KB (447 words) - 14:48, 5 June 2017
...tion, and $x$ is called the induction variable, induction parameter or the variable with respect to which the induction is carried out (in those cases when $P(
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===One real variable===
===One complex variable===
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''object variable''
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...field of coefficients. The absolutely irreducible polynomials of a single variable are the polynomials of degree 1. In the case of several variables there are
...ors. Over the field of real numbers any irreducible polynomial in a single variable is of degree 1 or 2 and a polynomial of degree 2 is irreducible if and only
3 KB (478 words) - 15:26, 30 December 2018
A set of random variables (cf. [[Random variable|Random variable]]) having finite expectations such that integrated tails of their distribut
converge in probability to a random variable $ X $(
3 KB (438 words) - 08:27, 6 June 2020
...and where it is stressed that only in this context the notion of a random variable becomes meaningful).
...n X(\omega)<x\}$ belongs to the class $\mathcal{A}$. Let $X$ be any random variable and $\mathcal{A}_X$ the class of subsets $C\subset\mathbf{R}^1$ for which $
6 KB (904 words) - 20:43, 1 January 2019
...em on the possibility of uniform approximation of functions of one complex variable by polynomials. Let $K$ be a compact subset of the complex $z$-plane $\math
..., "Sur la réprésentation par des séries de polynômes des fonctions d'une variable complexe dans des domaines fermés" ''Mat. Sb.'' , '''16''' : 3 (1945)
5 KB (707 words) - 19:02, 1 May 2014
''in the theory of functions of a complex variable''
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Sect. 86 (Translated from Russian)</TD><
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The problem of determining in a domain $D$ of the variable $x = (x_1,\ldots,x_n)$ the solution $u(x)$ of a linear differential equatio
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...on to functions of two variables of the [[Variation of a function]] of one variable, proposed by Tonelli in {{Cite|To}}.
...e $TV (g)$ denotes the classical total variation of a function of one real variable.
3 KB (386 words) - 10:54, 16 March 2023
of a complex variable. In the case of one complex variable $ z = x + iy $,
is an analytic function of the variable $ z = x _ {1} + ix _ {2} $.
4 KB (617 words) - 17:46, 4 June 2020
is an analytic function of the complex variable $ z $
of the complex variable $ z = ( z _ {1} \dots z _ {n} ) $,
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There is also a multi-variable generalization: generalized Bernstein polynomials defined by the completely
...variable case these provide explicit polynomial approximants for the more-variable Weierstrass approximation and Stone–Weierstrass theorems. For the behavio
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...[sample space]] with the property that if the observed value of a [[random variable]], the distribution of which is connected with the tested hypothesis, falls
be the tested hypothesis concerning the distribution of a random variable $ X $,
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...een domains of Euclidean spaces $\mathbf R^n$, $n\geq2$; when changing the variable in multiple integrals; when clarifying conditions for determining an implic
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an integer) of a random variable $ X $
is the [[Distribution function|distribution function]] of the random variable $ X $,
7 KB (1,019 words) - 08:53, 21 January 2024
in the case of discrete time). A [[Random variable|random variable]] $ \tau = \tau ( \omega ) $
3) The random variable
4 KB (526 words) - 07:59, 6 June 2020
...ero or is strictly positive and $\log f$ is a [[convex function (of a real variable)]].
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...ut to be a theory analogous to that of almost-periodic functions of a real variable (cf. [[Almost-periodic function|almost-periodic function]]). Therefore, man
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in one complex variable $z$ with positive radius of convergence $R$, $0<R\leq\infty$, one has in mi
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..."elementary translation" is also used: as are "algebraic function" (of one variable) or "polynomial".
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...se that a certain hypothesis $H_0$ concerning the distribution of a random variable $X$ is being tested, using a test based on a statistic $T(X)$ the distribut
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...Ref|Ca}}||valign="top"| C. Caratheodory, "Theory of functions of a complex variable", '''1''', Chelsea (1983) Sects. 274–275
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...o functions of several variables of the [[Variation of a function]] of one variable, proposed by
...[Function of bounded variation|function of bounded variation]] of one real variable. If a function has bounded Hardy variation, then it also has necessarily bo
4 KB (644 words) - 10:52, 10 December 2012
...e unknown function and $f ( x )$ is a known real-valued function of a real variable $x$. I.e. one asks for the eigenvalues and eigenfunctions of the compositio
...a3]</td> <td valign="top"> M. Kuczma, "Functional equations in a single variable" , PWN (1968)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> G
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...t|essential singular point]] of an analytic function $f(z)$ of the complex variable $z$, then there exists at least one ray $S=\{z:\arg(z-a)=\theta_0\}$ issuin
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. 345 (Translated from Russian)</TD></TR><
2 KB (343 words) - 21:49, 22 December 2018
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
...eins, "Selected topics in the classical theory of functions of a complex variable" , Holt, Rinehart & Winston (1962)</TD></TR>
2 KB (312 words) - 17:31, 13 November 2016
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)</TD><
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of a complex variable $ z $,
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...ons based on replacing them by equations involving functions of a discrete variable. See [[Hyperbolic partial differential equation, numerical methods|Hyperbol
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A complex variable $ t $
...TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian)</TD></TR>
5 KB (620 words) - 12:06, 9 December 2023
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
1 KB (245 words) - 08:26, 28 April 2023
...espect to $t$ of the function $y=f(t,u_1,\dotsc,u_m)$ which depends on the variable $t$ not only directly but also via the intermediate variables $u_1=u_1(t,x_
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...actor in front of a power of an unknown or a constant factor multiplying a variable quantity. Thus, the coefficient in the monomial $-(3/4)a^2b^2$ is $-3/4$; i
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...2}$ is the [[Median (in statistics)|median (in statistics)]] of the random variable $X$. The quantiles $K_{1/4}$ and $K_{3/4}$ are called the quartiles, and $K
2 KB (269 words) - 16:40, 5 August 2014
then the random variable $ \sqrt {X _ {1} ^ {2} + X _ {2} ^ {2} + X _ {3} ^ {2} } $
coincides with the distribution of the square root of a variable having the $ \chi ^ {2} $-
3 KB (351 words) - 11:10, 9 April 2023
...the analysis of models of the dependence of the mean value of some random variable $ Y $
...h random and non-random ones (controlled in the experiment); if the random variable $ Y $
4 KB (576 words) - 06:25, 18 January 2022
is a holomorphic function of the complex variable $ \xi $
of a complex variable $ z $
6 KB (901 words) - 16:08, 1 April 2020
...n in the theory of [[Analytic function|analytic functions]] of one complex variable and sometimes in the theory of [[Partial differential equation|partial diff
...="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a complex variable" , '''1–3''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}}
2 KB (344 words) - 08:39, 15 January 2014
...opediaofmath.org/legacyimages/c/c026/c026050/c0260502.png" />, to a random
variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
2 KB (315 words) - 16:40, 6 February 2012
...e (cf. [[Functions of a complex variable, theory of|Functions of a complex variable, theory of]]). It has its roots in quaternionic analysis, which was develop
The power of classical function theory of one complex variable is based on the Cauchy integral formula (cf. [[Cauchy integral theorem|Cauc
9 KB (1,297 words) - 17:44, 4 June 2020
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) pp. App
...of functions of a complex variable|Approximation of functions of a complex variable]].
6 KB (963 words) - 14:03, 18 May 2023
of a real variable is defined as the limit
of a real variable has a symmetric derivative of order $ 2r $
3 KB (376 words) - 08:24, 6 June 2020
A [[random variable]] taking values in the set $\Sigma_n$ of all single-valued mappings of $X$
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A measure of the linear dependence between one random variable and a certain collection of random variables. More precisely, if $ ( X _
the variable $ X _ {1} ^ {*} $
5 KB (666 words) - 08:02, 6 June 2020
...dule]]). Examples are the ring of integers, the ring of polynomials in one variable over a field, von Neumann regular rings (cf. [[Regular ring (in the sense o
...$2$-generated ideals of $R$ are projective. The ring of polynomials in one variable over a commutative ring $T$ is semi-hereditary if and only if $R$ is regula
3 KB (403 words) - 17:47, 27 August 2014
on the basis of a realization of a random variable $ X $
when the random variable $ X $
3 KB (411 words) - 08:11, 6 June 2020
...s variable) after substitution of which at all places where the considered variable appears in (*) one obtains the words $ Q _ {1} \dots Q _ {m} $
is a variable. Further details are contained in [[#References|[a1]]].
4 KB (601 words) - 14:35, 17 March 2023
(provided the variable $a$ does not occur in $A(x)$ and $C$).
713 bytes (113 words) - 08:11, 12 August 2014
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)</TD><
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...rdered spaces that are analogues of concave and convex functions of a real variable.
with respect to the variable $ u $.
2 KB (368 words) - 17:46, 4 June 2020
is a [[Random variable|random variable]] $ \xi $
...econd — of a discrete-time source of information. In other cases, a random variable $ \xi = \{ {\xi ( t) } : {t \in \Delta } \} $
8 KB (1,081 words) - 22:12, 5 June 2020
The dependence between variable quantities expressed by the logarithmic function was first considered by J.
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR>
6 KB (901 words) - 12:10, 1 May 2023
''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
with respect to the variable $ x _ {1} $
in the single variable $ x _ {1} $.
3 KB (454 words) - 07:57, 13 May 2022
...arzian differential parameter, of an analytic function $f(z)$ of a complex variable $z$''
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
3 KB (443 words) - 11:10, 30 December 2018
...artan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995).
..."top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)
2 KB (382 words) - 13:06, 10 December 2013
...sfiability, which defines the truth value of a formula relative to a given variable [[Assignement|assignment]].
...$ is used as a bounded variable in $p$ and if $p$ does not contain another variable $y\in X$, then for an assignement $v\in B(A,X)$ it holds {{Cite|TT95}} $$A\
8 KB (1,390 words) - 18:36, 11 December 2020
...d supercritical: the "criticality parameter" here (see {{Cite|AN}}) is the variable
...g process in a random medium is said to be subcritical and, for the random variable
5 KB (646 words) - 06:29, 30 May 2020
is a random variable with distribution function $ F ( x, t) $,
then the variable $ Z = Z ( X) = \Phi ^ {-1} [ F ( X , t) ] $
4 KB (608 words) - 16:41, 15 January 2021
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR></table>
2 KB (283 words) - 12:43, 14 February 2020
The function of a real variable $x$ which is equal to $1$ if $x$ is positive, equal to 0 if $x$ is zero and
630 bytes (102 words) - 15:41, 23 November 2012
...ny instruction can be labelled. A label in Snobol can be treated as a line variable whose value is an instruction labelled by this label. Fundamental data type
...nes is chosen in the subject. These sublines can be conferred by the above variable or replaced by other sublines.
6 KB (915 words) - 08:14, 6 June 2020
is a [[Convex function (of a real variable)|convex function (of a real variable)]] of the parameters $ \alpha $
and it is a convex function (of a real variable) in the domain $ 0 \leq \alpha , \beta \leq 1 $,
4 KB (618 words) - 13:00, 6 January 2024
in the fist variable and of the form $ C ^ \prime \rightarrow C \rightarrow C ^ {\prime\pr
Derived functors may also be defined for additive functors of a single variable between module categories, or, more generally, between arbitrary Abelian ca
3 KB (470 words) - 07:09, 10 May 2022
...in where a problem is to be numerically solved (see [[Variable-grid method|Variable-grid method]]). It is assumed that the dynamic parameters of the gas within
3 KB (431 words) - 10:52, 16 April 2014
''real-
variable theory of, real-
variable $\mathcal{H}^p$ theory''
The real-
variable Hardy spaces $\mathcal{H}^p = \mathcal{H}^p (\mathbb R^n)$ ($0<p<\infty$) a
6 KB (1,018 words) - 13:36, 26 December 2013
...he domain of validity of $P(x)$ is the same as the domain of values of the variable $x$. The statement $\exists x\ P(x)$ means that the domain of validity of $
...et-up can be generalized to polyadic "quantifiers" binding more than one variable occurring in more than one formula (example: the equi-cardinality quantifie
3 KB (468 words) - 11:33, 30 December 2018
...es of polynomials and best approximation of continuous functions of a real variable" , '''1''' , Moscow-Leningrad (1937) (In Russian)</TD></TR><TR><TD valign
...lign="top"> A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) pp. Chapt. 4 (Translated from Russian)</TD></TR><TR><
3 KB (397 words) - 20:25, 10 January 2021
Suppose that, based on a realization of a random variable $ X $
is a random variable taking values in a sample space $ ( \Theta , \mathfrak B _ \Theta , \pi
5 KB (695 words) - 22:16, 5 June 2020
More precisely, if for a function $ f $ of a real variable,
890 bytes (134 words) - 10:40, 20 January 2024
...ing media. Let the wave process be described by the [[wave equation]] with variable coefficients:
2 KB (313 words) - 12:00, 24 March 2024
...on it is possible to introduce the neutron flow function as an independent variable and employ the neutron flow approximation to obtain the neutron flow equati
...y value in unit time. The neutron flow function, which acts as independent variable $\tau$, is understood to mean the symbolic age of the neutrons, which is eq
3 KB (514 words) - 11:47, 5 July 2014
...cept of computability just outlined can be extended to functions of a real
variable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.
...clopediaofmath.org/legacyimages/b/b015/b015170/b01517021.png" /> of a real
variable is said to be computable according to Banach–Mazur (the set of such funct
7 KB (942 words) - 18:50, 24 March 2012
[[Category:Functions of a complex variable]]
1 KB (159 words) - 10:51, 18 October 2014
...tein, "Sur la définition et les propriétés des fonctions analytiques d'une variable réelle" ''Math. Ann.'' , '''75''' (1914) pp. 449–468</td></tr>
...s extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle" , Gauthier-Villars (1926)</td></tr>
4 KB (582 words) - 12:38, 23 February 2024
are integers. Analytic functions of one complex variable with more than two primitive periods do not exist, except for constants.
...TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) {{MR|044
4 KB (565 words) - 19:36, 5 June 2020
...t be expressed in terms of known functions. Quadrature (in the case of one variable) and cubature (for calculating multiple integrals) formulas have been deriv
1 KB (164 words) - 08:00, 16 April 2023
...Osculation]]). The osculating sphere can also be defined as the limit of a variable sphere passing through four points of $l$ as these points approach $M$. If
950 bytes (153 words) - 13:20, 29 April 2014
...f a real variable $t$, $0<t<\infty$, giving a function $F(p)$ of a complex variable $p$. It was considered by P. Laplace at the end of the eighteenth and begin
1 KB (206 words) - 12:31, 18 August 2014
of a random variable $ \xi $
dimensional random variable $ \xi $
3 KB (448 words) - 17:33, 5 June 2020
The function of a real variable that assigns to a real number $x$ the largest integer $\leq x$. The modern
1 KB (158 words) - 12:37, 27 August 2014
and with the following input–output operators. The variable output $ {\mathcal H} ( \xi _ {0} ) x ( t ) \equiv {\mathcal H} ( \xi _ {
and the corresponding variable internal state is defined by
4 KB (572 words) - 22:11, 5 June 2020
...r this class of equations. For example, an elliptic operator $L(D,x)$ with variable coefficients, written in the form
922 bytes (151 words) - 17:23, 26 September 2014
A random variable with uniform distribution on $ [0,\ 1] $
In statistical applications the procedure for constructing a random variable $ X $
6 KB (911 words) - 12:00, 22 December 2019
is the [[Entropy|entropy]] of the random variable $ \xi _ {k} ^ {n} = ( \xi _ {k} \dots \xi _ {n} ) $.
The variable $ \overline{H}\; ( u) $,
3 KB (447 words) - 22:12, 5 June 2020
A generalization of the concept of a [[Random variable|random variable]]. The term "random element" was coined by M. Fréchet [[#References|[1]]
for example, is reminiscent of the definition of a random variable. Let $ ( \Omega , {\mathcal A} , {\mathsf P} ) $
5 KB (804 words) - 08:09, 6 June 2020
A method in the theory of functions of a complex variable which is employed in solving extremal problems in classes of univalent and
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
6 KB (915 words) - 17:24, 7 February 2011
...nstruction of difference schemes for parabolic differential equations with variable coefficients the following methods are employed: balance, variational and f
...tion of systems of parabolic differential equations containing one spatial variable one uses the same difference schemes as for a single equation. The solution
18 KB (2,500 words) - 06:21, 12 October 2023
is a complex variable, and the
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR>
3 KB (470 words) - 08:17, 26 March 2023
==Trigonometric functions of a complex variable.==
The trigonometric functions for complex values of the variable $ z = x + iy $
12 KB (1,667 words) - 08:26, 6 June 2020
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
...the Weierstrass form of the Lamé equation. By a change of the independent variable in \eqref{eq1} one obtains Jacobi's form of the Lamé equation:
...tion to which is carried out by various transformations of the independent variable in \eqref{eq1}, for example:
3 KB (443 words) - 18:25, 1 May 2024
is a random variable with given a priori distribution $ \pi = \pi (d \theta ) $,
expresses the deviation between the variable $ \theta $
4 KB (497 words) - 10:33, 29 May 2020
''for measurability of a function of a real variable''
...2]</TD> <TD valign="top"> I.P. Natanson, "Theory of functions of a real variable" , '''1–2''' , F. Ungar (1955–1961) (Translated from Russian) {{MR|0
3 KB (436 words) - 19:43, 28 December 2017
is singular for the Emden equation. By the change of variable $ x = 1 / \xi $
and by the change of variable $ y = \eta / x $,
4 KB (508 words) - 19:37, 5 June 2020
be a random variable taking values in a sample space $ ( \mathfrak X , \mathfrak B , {\mathsf
with boundaries that are functions of the random variable $ X $
6 KB (853 words) - 22:13, 5 June 2020
...etween two random variables $X$ and $Y$ (cf. also [[Random variable|Random variable]]) with means $\mu_x=\mathsf{E}(X)$, $\mu_y=\mathsf{E}(Y)$ and finite varia
4 KB (683 words) - 07:24, 24 March 2023
is a random variable $ {\mathsf P} ( A \mid \mathfrak B ) $,
The conditional probability with respect to a random variable $ X $
4 KB (559 words) - 17:46, 4 June 2020
A random variable $ X $
...unimodal distribution and has positive asymmetry. The moments of a random variable $ X $
4 KB (482 words) - 22:17, 5 June 2020
An important class of real-valued continuous functions of a single
variable consists of those functions that are continuous on intervals. They have the
...ediaofmath.org/legacyimages/c/c025/c025650/c025650117.png" /> in, say, the
variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
26 KB (3,622 words) - 17:12, 7 February 2011
* A single term $t\in T(\Sigma,X)$ containing a variable $x\in X_s$ of sort $s\in S$ can be used for representing the infinite colle
...manipulation). In this case, the term $t$ with subterm $t_1$ replaced by a variable $x\in X_s$ not contained in $t$ defines the so-called context of the manipu
5 KB (955 words) - 16:09, 28 January 2013
...of functions of a complex variable|Approximation of functions of a complex variable]]). Then $R ( X )$ is a hypo-Dirichlet algebra [[#References|[a3]]].
3 KB (474 words) - 19:56, 27 February 2021
are also Bernoulli equations if $y$ is considered as the independent variable, while $x$ is an unknown function of $y$.
1 KB (175 words) - 18:48, 9 January 2024
...of functions of a complex variable|approximation of functions of a complex variable]]. It contains a section on Faber expansions.
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 3.14 (Translated from Russian)</T
5 KB (711 words) - 19:38, 5 June 2020
(linear with respect to the observed values of the random variable $ \mathbf Y $
of the resulting test (random variable) under investigation, $ \mathbf X $
3 KB (458 words) - 22:17, 5 June 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
dimensional random variable, $ Z $
In the case of a single endogenous variable, $ y = a ^ {T} x $,
3 KB (484 words) - 18:22, 18 December 2020
...a12]</td> <td valign="top"> J. Werner, "Über die globale konvergenz von Variable-Metric Verfahren mit nichtexakter Schrittweitenbestimmung" ''Numer. Math.'
13 KB (1,868 words) - 07:21, 13 February 2024
...\mathbb C$ is holomorphic at every point $\zeta\in D$ with respect to each variable $z_k$, then it is holomorphic with respect to all variables.
In the theorem above ''holomorphic with respect to the variable'' $z_k$ means that, if $\zeta = (\zeta_1, \ldots, \zeta_n)$, then the map
6 KB (1,001 words) - 18:22, 31 March 2017
...the solution of the Cauchy problem for the wave equation with one spatial variable. Let the given functions $ \phi $,
3 KB (438 words) - 13:28, 26 March 2023
...te fibres play an important role in the theory of functions of one complex variable: these include all holomorphic functions in a domain. The theorem on the op
...TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD
3 KB (511 words) - 08:04, 6 June 2020
...lign="top"> A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)</TD></TR><TR><TD valign="top
6 KB (901 words) - 17:32, 5 June 2020
...of information contained in one random variable relative to another random variable. Let $ \xi $
4 KB (517 words) - 16:27, 6 January 2024
of the complex variable $ z = x _ {1} + ix _ {2} $
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><TD va
3 KB (480 words) - 18:31, 5 April 2020
The Luzin–Privalov theorems in the theory of functions of a complex variable are classical results of N.N. Luzin and I.I. Privalov that clarify the char
1) Let $f(z)$ be a meromorphic function of the complex variable $z$ in a simply-connected domain $D$ with rectifiable boundary $\Gamma$. If
3 KB (424 words) - 21:56, 24 July 2012
...visual advantage that neighbouring cells are represented by adjacent input variable combinations, i.e., by binary numbers that differ in only one bit position.
...re independent. However, the map can be readily modified to handle certain variable dependencies (such as certain cell mutual exclusions), which results in red
9 KB (1,380 words) - 22:14, 5 June 2020
of a real variable $ x $:
1,017 bytes (148 words) - 08:04, 6 June 2020
A [[Probability distribution|probability distribution]] of a random variable $ X $
is a random variable having, conditionally on $ \lambda $,
4 KB (583 words) - 08:02, 6 June 2020
...othesis being tested and under all possible states of nature. Let a random variable $ X $
Suppose it is necessary, via a realization of the random variable $ X $,
4 KB (508 words) - 22:16, 5 June 2020
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
...="top">[a2]</TD> <TD valign="top"> J.B. Conway, "Functions of a complex variable" , Springer (1978)</TD></TR></table>
7 KB (959 words) - 22:14, 5 June 2020
of the complex variable $z$. It is important for its applications in fluid mechanics, which were di
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><T
4 KB (542 words) - 19:58, 4 January 2024
..."top"> S.N. Mergelyan, "Uniform approximation to functions of a complex variable" ''Transl. Amer. Math. Soc.'' , '''3''' (1962) pp. 294–391 ''Uspekhi
2 KB (243 words) - 18:27, 13 August 2015
...lgebraically closed. Further examples are given by function fields in one variable over algebraically closed fields: this is Tsen's theorem. Chevalley proved
1 KB (152 words) - 19:37, 17 November 2023
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)</TD><
1 KB (182 words) - 19:25, 13 December 2015
...h $M$ characterized by the property that the distance from this plane to a variable point $M_1$ of $S$ as $M_1$ approaches arbitrarily close to $M$ is infinite
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===Functions of one real variable===
...ncept of semicontinuous function was first introduced for functions of one variable, using [[Upper and lower limits|upper and lower limits]].
8 KB (1,260 words) - 15:29, 5 January 2017
denotes the mathematical expectation of a random variable $ X $,
1 KB (159 words) - 18:48, 5 April 2020
...ers of a given distribution. More precisely, for a realization of a random variable $ X $,
2 KB (341 words) - 08:03, 6 June 2020
''of a random variable $X$''
The function of a real variable $x$ taking at each $x$ the value equal to the probability of the inequality
6 KB (864 words) - 13:51, 12 December 2013
is a real-valued function, defined, respectively, for a real variable and an integer argument.
....S. Besicovitch, "On mean values of functions of a complex and of a real variable" ''Proc. London Math. Soc. (2)'' , '''27''' (1927) pp. 373–388</TD></T
4 KB (506 words) - 14:27, 30 May 2020
of the hypercomplex variable $ z $(
...neralizations in different directions of analytic functions of one complex variable. The concepts of an analytic hypercomplex function differ, on account of th
5 KB (724 words) - 22:11, 5 June 2020
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian)</TD></TR>
2 KB (238 words) - 21:25, 13 December 2016
..."top"> Mourad Ismail, "Classical and Quantum Orthogonal Polynomials in One Variable", Encyclopedia of mathematics and its applications '''98''' , Cambridge Uni
1 KB (188 words) - 20:27, 15 November 2023
..."chi-squared" distribution. For example, the distribution of the random variable $ \sqrt {\chi _ {n} ^ {2} } $—
...according to the central limit theorem, the distribution of the normalized variable $ ( \chi _ {n} ^ {2} - n)/ \sqrt 2n $
4 KB (555 words) - 16:43, 4 June 2020
...ets (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]; [[Convex functional|Convex functional]]; [[Convex set|Convex set]]). Th
4 KB (527 words) - 17:15, 7 February 2011
...bouring regions have different colours. To each region he associated a $2$-variable monomial $\delta ^ { i } \lambda ^ { j }$, where $i$ is the number of $\del
...n has corners of the same colour, and that his polynomials are in fact one-variable polynomials. He also stated that the left- and right-handed trefoil knots a
3 KB (496 words) - 07:37, 18 March 2023
.../TD> <TD valign="top"> P.S. Aleksandrov, "Theory of functions of a real variable and the theory of topological spaces" , Moscow (1978) (In Russian)</TD></
1 KB (203 words) - 17:25, 1 January 2021
...n. Suppose, for example, that there are reasons for assuming that a random variable $ Y $
of another variable, so that
12 KB (1,774 words) - 13:04, 13 January 2024
...h]] (cf. [[Convex function (of a real
variable)|Convex function (of a real
variable)]]). The convexity of a functional <img align="absmiddle" border="0" src="h
...f. [[Convex function (of a complex
variable)|Convex function (of a complex
variable)]]).
15 KB (2,101 words) - 17:13, 7 February 2011
..._{ij}$, $i=1,\ldots,n$; $j = 1,\ldots,m_i$, is either an atomic formula (a variable or constant) or the negation of an atomic formula. The conjunctive normal f
938 bytes (159 words) - 09:28, 29 November 2014
...example, the modulus $|f(z)|$ of an analytic function $f(z)$ of a complex variable is a logarithmically-subharmonic function, but there are continuous logarit
1 KB (205 words) - 15:22, 29 December 2018
...ethods of solving problems of minimization of [[Convex function (of a real variable)|convex functions]] on [[convex set]]s defined by systems of equalities and
2 KB (236 words) - 12:57, 3 May 2016
Similar assertions are also valid for functions of a complex variable $ z $
in a neighbourhood of the point at infinity or inside an angle. For a complex variable 5) takes the following form: If the function $ f(z) $
4 KB (542 words) - 06:35, 14 April 2024
see [[Convex function (of a real variable)|Convex function (of a real variable)]]), $ x _ {i} \in C $,
4 KB (524 words) - 22:14, 5 June 2020
which is the so-called Gaussian probability integral. For a random variable $ X $
...tives are illustrated in the figure. Regarded as a function of the complex variable $ z $,
4 KB (598 words) - 19:23, 11 January 2024
...the same way as the Lebesgue integral in the general case. If $\mu$ is of variable sign, then $\mu=\mu_1-\mu_2$, where $\mu_1$ and $\mu_2$ are non-negative me
1 KB (208 words) - 13:07, 26 August 2014
A term in [[Discriminant analysis|discriminant analysis]] denoting a variable used to establish a rule for assigning an object with measurements $x=(x_1,
1 KB (185 words) - 12:24, 14 February 2020
...ts to test whether the distribution function of a [[Random variable|random variable]] $ X $
5 KB (709 words) - 06:49, 26 March 2023
...ere must be syntactic rules of formation of sorts, also called types. Each variable of such a language has a definite type. The semantic rules must associate w
...ameters, that is, have free entries of variables (see [[Free variable|Free variable]]). From the semantic point of view a term without parameters (a closed ter
9 KB (1,407 words) - 19:01, 21 November 2018
valued random variables (cf. [[Random variable|Random variable]]) converges in distribution if and only if their finite-dimensional distri
4 KB (518 words) - 08:14, 6 June 2020
...opediaofmath.org/legacyimages/a/a013/a013980/a0139807.png" /> of a complex
variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
5 KB (730 words) - 17:11, 7 February 2011
...ields: finite fields, local fields or fields of algebraic functions in one variable.
...over the base field $\mathbf C(X)$, the field of rational functions in one variable over the complex numbers, which is well under control. Under certain condit
4 KB (586 words) - 04:07, 25 February 2022
...elements]]). A theory has been developed of boundary correspondence under variable conformal mappings of simply-connected domains (see [[#References|[3]]]).
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1965–1977) (Translated from Russian)</TD></TR><TR
4 KB (564 words) - 21:51, 16 December 2020
An information-theoretical measure of the degree of indeterminacy of a random variable. If $ \xi $
is a discrete random variable defined on a probability space $ ( \Omega , \mathfrak A , {\mathsf P} ) $
7 KB (1,067 words) - 13:06, 6 January 2024
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><TD va
2 KB (301 words) - 08:09, 6 June 2020
that are entire functions of the variable $w_n=1/(z-a_n)$, there is a single-valued analytic function $f(z)$ having s
...TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|
6 KB (980 words) - 18:47, 24 May 2017
...random variables with the same distribution (cf. [[Random variable|Random variable]]), and let $ S _ {0} = 0 $,
4 KB (549 words) - 18:47, 5 April 2020
A coefficient of an independent variable in a [[Regression|regression]] equation. For example, in the linear regress
1 KB (146 words) - 08:10, 6 June 2020
...n mathematical analysis. Suppose that a real-valued function $f$ of a real variable $x$ is defined in a neighborhood of a point $x_0$ and that there exists a f
...change to complex-valued and vector-valued functions of a real or complex variable. Moreover, there exists a concept of a derivative of a scalar-valued point
4 KB (596 words) - 11:47, 5 July 2016
...arities]] of [[Holomorphic function|holomorphic functions]] of one complex variable
...="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a complex variable" , '''1–3''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}}
3 KB (465 words) - 20:00, 14 December 2014
A method in the theory of functions of a complex variable that is based on inequalities relating the lengths of curves of a certain s
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
4 KB (542 words) - 16:52, 19 February 2022
...|Osculation]]). The osculating plane can also be defined as the limit of a variable plane passing through three points of $l$ as these points approach $M$. Usu
1 KB (233 words) - 16:51, 8 April 2023
...tilde D \rightarrow D$, just as for every analytic function of one complex variable there is a corresponding [[Riemann surface]]; the function $f$ is single-va
1 KB (205 words) - 17:02, 13 June 2020