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31: 3.4 Differentiation

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3.4.17 $$\frac{1}{k!}{f}^{(k)}(x_0)=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{{(\zeta -x_0)}^{k+1}}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $C$: simple closed contour

Permalink:

http://dlmf.nist.gov/3.4.E17

Encodings:

pMML, png, TeX

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Annotations for §3.4(ii), §3.4 and Ch.3

►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. …

32: 10.74 Methods of Computation

… ►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). …

33: 25.15 Dirichlet $L$-functions

… ►For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). …

34: 3.7 Ordinary Differential Equations

… ►For applications to special functions $f$, $g$, and $h$ are often simple rational functions. … ►The eigenvalues ${\lambda }_k$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …

35: 22.4 Periods, Poles, and Zeros

… ►Then: (a) In any lattice unit cell $\mathrm{p}\mathrm{q}(z,k)$ has a simple zero at $z=\text{p}$ and a simple pole at $z=\text{q}$. …

36: 23.2 Definitions and Periodic Properties

… ►The poles of $\mathrm{\wp }\left(z\right)$ are double with residue $0$; the poles of $\zeta \left(z\right)$ are simple with residue $1$. The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. …

37: 29.12 Definitions

… ►The polynomial $P(\xi )$ is of degree $n$ and has $m$ zeros (all simple) in $(0,1)$ and $n-m$ zeros (all simple) in $(1,k^{-2})$. …

38: Bibliography R

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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39: 2.10 Sums and Sequences

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2.10.26 $$f_n=\frac{1}{2\pi \mathrm{i}}{\int }_{𝒞}\frac{f(z)}{z^{n+1}}dz,$$

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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, $f(x)$: analytic function, $f_n$: coefficients and $𝒞$: simple closed contour

Permalink:

http://dlmf.nist.gov/2.10.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

►where $𝒞$ is a simple closed contour in the annulus that encloses $z=0$. …

40: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For $T$ to be actually self adjoint it is necessary to also show that $𝒟(T^{\ast })=𝒟(T)$, as it is often the case that $T$ and $T^{\ast }$ have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator $\frac{d}{dx}$. ►This question may be rephrased by asking: do $f(x)$ and $g(x)$ satisfy the same boundary conditions which are needed to fully specify the solutions of a second order linear differential equation? A simple example is the choice $f(a)=f(b)=0$, and $g(a)=g(b)=0$, this being only one of many. … ►

Example 1: Three Simple Cases where $q(x)=0$, $X=[0,\pi ]$

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§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

… ►this being a matrix element of the resolvent $F(T)={\left(z-T\right)}^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). …