# simply-connected domain

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## 1—10 of 42 matching pages

##### 1: 1.13 Differential Equations

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###### §1.13(i) Existence of Solutions

►A domain in the complex plane is*simply-connected*if it has no “holes”; more precisely, if its complement in the extended plane $\u2102\cup \{\mathrm{\infty}\}$ is connected. … ►where $z\in D$, a simply-connected domain, and $f(z)$, $g(z)$ are analytic in $D$, has an infinite number of analytic solutions in $D$. … ►
1.13.6
$$A{w}_{1}(z)+B{w}_{2}(z)=0,$$
$\forall z\in D$,

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$u$ and $z$ belong to domains
$U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$).
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##### 2: 3.3 Interpolation

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►If $f$ is analytic in a simply-connected domain
$D$ (§1.13(i)), then for $z\in D$,
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►If $f$ is analytic in a simply-connected domain
$D$, then for $z\in D$,
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##### 3: Bibliography Y

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables.
In Hypergeometric Functions on Domains of Positivity, Jack
Polynomials, and Applications (Tampa, FL, 1991),
Contemporary Mathematics, Vol. 138, pp. 239–259.
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The calculation of the Riemann zeta function in the complex domain.
USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.
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Computation of the derivatives of the Riemann zeta-function in the complex domain.
USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.
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##### 4: Possible Errors in DLMF

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►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula.
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##### 5: 14.26 Uniform Asymptotic Expansions

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►The uniform asymptotic approximations given in §14.15 for ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(x\right)$ for $$ are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986).
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##### 6: Donald St. P. Richards

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►He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis

*: A Conference in Honor of R. A. Kunze*(with T. …##### 7: Bonita V. Saunders

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►As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains.
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##### 8: 1.10 Functions of a Complex Variable

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►Let ${f}_{1}(z)$ be analytic in a domain
${D}_{1}$.
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►Let $F(z)$ be a multivalued function and $D$ be a domain.
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►A

*cut domain*is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … … ►Suppose $D$ is a domain, and …##### 9: 1.9 Calculus of a Complex Variable

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►When its boundary points are added the domain is said to be

*closed*, but unless specified otherwise a domain is assumed to be open. … ► … ►###### Jordan Curve Theorem

… ►One of these domains is bounded and is called the*interior domain of*$C$; the other is unbounded and is called the*exterior domain of*$C$. …##### 10: 16.23 Mathematical Applications

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►The Bieberbach conjecture states that if ${\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}{z}^{n}$ is a conformal map of the unit disk to any complex domain, then $|{a}_{n}|\le n|{a}_{1}|$.
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