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21: 1.6 Vectors and Vector-Valued Functions

… ►The geometrical image $C$ of a path $𝐜$ is called a simple closed curve if $𝐜$ is one-to-one, with the exception $𝐜(a)=𝐜(b)$. … … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. …

22: 3.3 Interpolation

… ►

3.3.6 $$R_n(z)=\frac{{\omega }_{n+1}(z)}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{(\zeta -z){\omega }_{n+1}(\zeta )}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $f(z)$: function, ${\omega }_{n+1}(z)$: nodal polynomial, $R_n(x)$: remainder and $C$: simple closed contour in $D$

A&S Ref:

25.2.5

Permalink:

http://dlmf.nist.gov/3.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(i), §3.3 and Ch.3

►where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,z_1,z_2,\mathrm{\dots },z_n$. … ►

3.3.37 $$\left[z_0,z_1,\mathrm{\dots },z_n\right]f=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{{\omega }_{n+1}(\zeta )}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\left[z_0,z_1,\mathrm{\dots },z_n\right]f$: divided difference, $\mathrm{i}$: imaginary unit, $\int$: integral, $f(z)$: function, $C$: simple closed contour in $D$ , $z_k$: nodes of function and ${\omega }_{n+1}(z)$: nodal polynomial

Permalink:

http://dlmf.nist.gov/3.3.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(iii), §3.3 and Ch.3

►where ${\omega }_{n+1}(\zeta )$ is given by (3.3.3), and $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing $z_0,z_1,\mathrm{\dots },z_n$. …

23: 20.13 Physical Applications

… ►Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). …

24: 28.7 Analytic Continuation of Eigenvalues

… ►The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …

25: 2.8 Differential Equations with a Parameter

… ►In Case II $f(z)$ has a simple zero at $z_0$ and $g(z)$ is analytic at $z_0$. … ►

§2.8(iii) Case II: Simple Turning Point

… ►

§2.8(iv) Case III: Simple Pole

… ►More generally, $g(z)$ can have a simple or double pole at $z_0$. … ►For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …

26: 7.13 Zeros

… ► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►At $z=0$, $C\left(z\right)$ has a simple zero and $S\left(z\right)$ has a triple zero. …

27: 18.40 Methods of Computation

… ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). … ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define …

28: 22.2 Definitions

… ►Each is meromorphic in $z$ for fixed $k$, with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$. …

29: 25.2 Definition and Expansions

… ►It is a meromorphic function whose only singularity in $ℂ$ is a simple pole at $s=1$, with residue 1. … ►

25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

…

30: 28.12 Definitions and Basic Properties

… ►When $q=0$ Equation (28.2.16) has simple roots, given by … ►As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. …