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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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External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

►

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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Notes:

Includes Fortran programs.

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i), §25.21(iii).

Encodings:

BibTeX

…

2: 3.7 Ordinary Differential Equations

… ►where $f$, $g$, and $h$ are analytic functions in a domain $D\subset ℂ$. … ►Assume that we wish to integrate (3.7.1) along a finite path $𝒫$ from $z=a$ to $z=b$ in a domain $D$. …

3: 1.10 Functions of a Complex Variable

… ►Let $f_1(z)$ be analytic in a domain $D_1$. … ►Suppose the subarc $z(t)$, $t\in [t_{j-1},t_j]$ is contained in a domain $D_j$, $j=1,\mathrm{\dots },n$. … ►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 $D$. … ►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. … ►Suppose $a_n=a_n(z)$, $z\in D$, a domain. …

4: 1.9 Calculus of a Complex Variable

… ►A domain $D$, say, is an open set in $ℂ$ 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

… ►A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane $ℂ\cup \{\mathrm{\infty }\}$ 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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Symbols:

$\in$: element of, $\forall$: for every, $z$: variable, $D$: simply-connected domain, $w_1(z)$: solution, $w_2(z)$: solution, $A$: constant and $B$: constant

Permalink:

http://dlmf.nist.gov/1.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

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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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External Links:

ISSN 1364-5021, Document, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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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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External Links:

ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt

Referenced by:

§32.11(i), §32.12(i).

Encodings:

BibTeX

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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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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(3), pp. 15–24

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(iv).

Encodings:

BibTeX

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8: 3.10 Continued Fractions

… ►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). …

9: 3.3 Interpolation

… ►If $f$ is analytic in a simply-connected domain $D$ (§1.13(i)), then for $z\in D$, … ►If $f$ is analytic in a simply-connected domain $D$, then for $z\in D$, …

10: 37.18 Orthogonal Polynomials on Quadratic Domains

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$${𝕍}^{d+1}=\{(x,t)\mid \Vert 𝐱\Vert \le |\varphi (t)|,t\in (a,b),x\in {ℝ}^d\},$$

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37.18.2 $$⟨f,g⟩=c_W{\int }_{{𝕍}^{d+1}}f(𝐱,t)g(𝐱,t)W(𝐱,t)dxdt,$$

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Defines:

$⟨f,g⟩$: inner product on quadratic domain (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}^{d+1}$: quadratic domain, $W$: weight function in $d$-variables and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(i), §37.18, Part 37.III and Ch.37

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37.18.8 $$[t(1-t)D_{tt}+2(1-t)⟨𝐱,{\nabla }_{𝐱}⟩D_t+t{\mathrm{\Delta }}_{𝐱}-{⟨𝐱,{\nabla }_{𝐱}⟩}^2+(2\mu +d)D_t-(2\mu +\gamma +d+1)(⟨𝐱,{\nabla }_{𝐱}⟩+t{D}_t)+⟨𝐱,{\nabla }_{𝐱}⟩]u=-n(n+2\mu +\gamma +d)u,$$ $\forall u\in {𝒱}_n({𝕍}^{d+1},W_{\mu ,0,\gamma }),d\ge 2,\mu >-\frac{1}{2},\gamma >-1$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\forall$: for every, $⟨𝐮,𝐯⟩$: inner product over vectors, $(a,b)$: open interval, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}^{d+1}$: quadratic domain and $W$: weight function in $d$-variables

Proved:

Xu (2020, Theorem 3.2)(proved)

Referenced by:

item 2

Permalink:

http://dlmf.nist.gov/37.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(ii), §37.18(ii), §37.18, Part 37.III and Ch.37

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37.18.13 $$[t({\mathrm{\Delta }}_{𝐱}+D_{tt})+2⟨𝐱,{\nabla }_{𝐱}⟩D_t-⟨𝐱,{\nabla }_{𝐱}⟩+(2\mu +d-t)D_t]u=-nu,$$ $\forall u\in {𝒱}_n({𝕍}^{d+1},W_{\mu ,0}),\mu >-\frac{1}{2}$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\forall$: for every, $⟨𝐮,𝐯⟩$: inner product over vectors, $(a,b)$: open interval, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕍}^{d+1}$: quadratic domain and $W$: weight function in $d$-variables

Proved:

Xu (2020, Theorem 7.3)(proved)

Referenced by:

item 5

Permalink:

http://dlmf.nist.gov/37.18.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(iii), §37.18(iii), §37.18, Part 37.III and Ch.37

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