# in a domain

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## 1—10 of 42 matching pages

##### 1: Bibliography Y

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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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##### 2: 3.7 Ordinary Differential Equations

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►where $f$, $g$, and $h$ are analytic functions in a domain
$D\subset \u2102$.
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►Assume that we wish to integrate (3.7.1) along a finite path $\mathcal{P}$ from $z=a$ to $z=b$
in a domain
$D$.
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##### 3: 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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►Suppose the subarc $z(t)$, $t\in [{t}_{j-1},{t}_{j}]$ is contained in a domain
${D}_{j}$, $j=1,\mathrm{\dots},n$.
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►If $f(z)$ is analytic in a domain
$D$, ${z}_{0}\in D$ and $\left|f(z)\right|\le \left|f({z}_{0})\right|$ for all $z\in D$, then $f(z)$ is a constant in
$D$.
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►Assume that for each $t\in [a,b]$, $f(z,t)$ is an analytic function of $z$
in
$D$, and also that $f(z,t)$ is a continuous function of both variables.
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►Suppose ${a}_{n}={a}_{n}(z)$, $z\in D$, a domain.
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##### 4: 1.9 Calculus of a Complex Variable

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►A

*domain*$D$, say, is an open set in $\u2102$ that is*connected*, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … ►A function $f(z)$ is*analytic in a domain*$D$ if it is analytic at each point of $D$. … … ►Suppose $f(z)$ is analytic in a domain $D$ and ${C}_{1},{C}_{2}$ are two arcs in $D$ passing through ${z}_{0}$. … ►Suppose the series ${\sum}_{n=0}^{\mathrm{\infty}}{f}_{n}(z)$, where ${f}_{n}(z)$ is continuous, converges uniformly on every*compact set*of a domain $D$, that is, every closed and bounded set in $D$. …##### 5: 1.13 Differential Equations

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►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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##### 6: Bibliography W

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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain.
Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.
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##### 7: Bibliography K

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Connection formulae for the first Painlevé transcendent in the complex domain.
Lett. Math. Phys. 27 (4), pp. 243–252.
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Calculation of modified Bessel functions in a complex domain.
Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.
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##### 8: 3.10 Continued Fractions

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►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5).
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##### 9: 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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##### 10: Bibliography H

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Development of a Gaussian hypergeometric function code in complex domains.
Internat. J. Modern Phys. C 4 (4), pp. 805–840.
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