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31: 8.11 Asymptotic Approximations and Expansions

… ►

8.11.2 $$\mathrm{\Gamma }(a,z)=z^{a-1}{\mathrm{e}}^{-z}\left(\sum _{k=0}^{n-1}\frac{u_k}{z^k}+R_n(a,z)\right),$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $n$: nonnegative integer, $u_k$ and $R_n(a,z)$: remainder

A&S Ref:

6.5.32

Permalink:

http://dlmf.nist.gov/8.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

… ►

8.11.4 $$\gamma (a,z)=z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{{\left(a\right)}_{k+1}},$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.33 (corrected)

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(ii), §8.11 and Ch.8

►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. … ►

8.11.6 $$\gamma (a,z)\sim -z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ , $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Notes:

See Paris (2002b)

Referenced by:

§8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, . It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where and $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

… ►

8.11.18 $$S_n(x)\sim \sum _{k=0}^{\mathrm{\infty }}{d}_k(x)n^{-k},$$ $n\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $S_n(x)$: function and $d_k(x)$: coefficients

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(v), §8.11 and Ch.8

…

32: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A (finite or countably infinite, generalizing the definition of (1.2.40)) set $\{v_n\}$ is an orthonormal set if the $v_n$ are normalized and pairwise orthogonal. … ►where the infinite sum means convergence in norm, … ►If ${sup}_{v\in 𝒟(T),\Vert v\Vert =1}\Vert Tv\Vert$ is finite then $T$ is bounded, and $T$ extends uniquely to a bounded linear operator on $V$. … ►Consider the second order differential operator acting on real functions of $x$ in the finite interval $[a,b]\subset ℝ$ … ►Let $X=(a,b)$ be a finite or infinite open interval in $ℝ$. …

33: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

►

33.6.2 $$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►

33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

… ►The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$. …

34: 1.10 Functions of a Complex Variable

… ►Next, $z_0$ is a pole if $a_n\ne 0$ for at least one, but only finitely many, negative $n$. … ►If $f(z)$ is analytic within a simple closed contour $C$, and continuous within and on $C$—except in both instances for a finite number of singularities within $C$—then … ►A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … ►Let $D$ be a domain and $[a,b]$ be a closed finite segment of the real axis. … ►(The integer $k$ may be greater than one to allow for a finite number of zero factors.) …

35: 18.39 Applications in the Physical Sciences

… ►where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\mathrm{\hslash }=h/(2\pi )$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval. … ►As in classical dynamics this sum is the total energy of the one particle system. … ►The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). … ►The fact that both the eigenvalues of (18.39.31) and the scaling of the $r$ co-ordinate in the eigenfunctions, (18.39.30), depend on the sum $p+l+1$ leads to the substitution … ►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional $L^2$ functions of (18.39.53), provided that such information involves potentials, or projections onto $L^2$ functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). …

36: 23.20 Mathematical Applications

… ►Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …Both $T$ and $I$ are finite sets. …To determine $T$, we make use of the fact that if $(x,y)\in T$ then $y^2$ must be a divisor of $\mathrm{\Delta }$; hence there are only a finite number of possibilities for $y$. …The resulting points are then tested for finite order as follows. …If any of these quantities is zero, then the point has finite order. …

37: 3.5 Quadrature

… ►

3.5.5 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=h\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}f(kh)+E_h(f),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $E_n(f)$: error term

Referenced by:

§3.5(i)

Permalink:

http://dlmf.nist.gov/3.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(i), §3.5 and Ch.3

… ►

3.5.12 $$G_0(\frac{1}{2}h)=\frac{1}{2}{G}_0(h)+\frac{1}{2}h\sum _{k=0}^{n-1}f\left(x_0+(k+\frac{1}{2})h\right),$$

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Symbols:

$G_k(h)$: Romberg scheme

Permalink:

http://dlmf.nist.gov/3.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(iii), §3.5 and Ch.3

… ►

3.5.15 $${\int }_a^{b}f(x)w(x)dx=\sum _{k=1}^n{w}_{k}f(x_k)+E_n(f),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $E_n(f)$: error term, $w$: weight and $w_k$: weights

Referenced by:

§3.5(v)

Permalink:

http://dlmf.nist.gov/3.5.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(iv), §3.5 and Ch.3

… ►Let $\{p_n\}$ denote the set of monic polynomials $p_n$ of degree $n$ (coefficient of $x^n$ equal to $1$) that are orthogonal with respect to a positive weight function $w$ on a finite or infinite interval $(a,b)$; compare §18.2(i). …In particular, with $h_m={\int }_a^b{p}_m{(x)}^{2}w(x)dx$, we have a finite system of orthogonal polynomials $p_m(x)$ ($m=0,1,\mathrm{\dots },n-1$) on $\{x_1,x_2,\mathrm{\dots },x_n\}$ with respect to the weights $w_k$: …

38: 1.4 Calculus of One Variable

… ►If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. … ►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha$ is absolutely continuous if $\alpha (x)$ is continuous and there exists a weight function $w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that … ►Similarly, assume that ${\int }_{-b}^{b}f(x)dx$ exists for all finite values of $b$ ($>0$), but not necessarily when $b=\mathrm{\infty }$. … ►With , the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is … ►Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. …

39: 15.17 Mathematical Applications

… ►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. … ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

40: 1.16 Distributions

… ►A sequence $\{{\varphi }_n\}$ of functions in $𝒯$ is said to converge to a function $\varphi \in 𝒯$ as $n\to \mathrm{\infty }$ if the sequence $\{{\varphi }_n^{(k)}\}$ converges uniformly to ${\varphi }^{(k)}$ on every finite interval and if the constants $c_{k,N}$ in the inequalities … ►

1.16.31 $$P(𝐱)=\sum _{𝜶}{c}_{𝜶}{𝐱}^{𝜶}=\sum _{𝜶}{c}_{𝜶}{x}_1^{{\alpha }_1}\mathrm{\cdots }{x}_n^{{\alpha }_n},$$

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Symbols:

$n$: nonnegative integer and $P$: polynomial of several variables

Permalink:

http://dlmf.nist.gov/1.16.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

… ►

1.16.32 $$P(𝐃)=\sum _{𝜶}{c}_{𝜶}{𝐃}^{\alpha }=\sum _{𝜶}{c}_{𝜶}{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\dots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

(1.16.30), (1.16.33), (1.16.34), (1.16.36), (1.16.37), item (iii), item (iii)

Permalink:

http://dlmf.nist.gov/1.16.E32

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was written

$$P(𝐷)=\sum _{𝜶}{c}_{𝜶}{𝐷}_{𝜶}$$

where ${𝐷}_{𝜶}$ was defined in (1.16.30). The change was made to clarify the meaning of this equation.

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

►Here $𝜶$ ranges over a finite set of multi-indices, $P(𝐱)$ is a multivariate polynomial, and $P(𝐃)$ is a partial differential operator. …