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41: 9.12 Scorer Functions

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9.12.15 $$\mathrm{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\cos \left(\frac{2k-1}{3}\pi \right)\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Substitute into (9.12.11) by means of (9.12.17), (9.4.3), (9.2.5), (9.2.6), and use (5.5.3).

Referenced by:

(9.12.16)

Permalink:

http://dlmf.nist.gov/9.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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9.12.17 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi }\sum _{k=0}^{\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{k+1}{3}\right)\frac{{(3^{1/3}z)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Proof sketch:

Expand the integral in (9.12.20) in powers of $zt$ and integrate term-by-term by means of (5.2.1).

Referenced by:

(9.12.15), (9.12.18)

Permalink:

http://dlmf.nist.gov/9.12.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vi), §9.12 and Ch.9

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42: 10.43 Integrals

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43: Bibliography D

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The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

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External Links:

ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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44: 3.5 Quadrature

… ►If $k$ in (3.5.4) is not arbitrarily large, and if odd-order derivatives of $f$ are known at the end points $a$ and $b$, then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula (§2.10(i)). … ►These can be found by means of the recursion … ►When $f\in C^{\mathrm{\infty }}$, the Romberg method affords a means of obtaining high accuracy in many cases with a relatively simple adaptive algorithm. …

45: 10.41 Asymptotic Expansions for Large Order

… ►To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively. …

46: 19.7 Connection Formulas

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47: 26.12 Plane Partitions

… ►Here $(h,j,k)\le (r,s,t)$ means $h\le r$, $j\le s$, and $k\le t$. …

48: 10.18 Modulus and Phase Functions

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49: 19.36 Methods of Computation

… ►All cases of $R_F$, $R_C$, $R_J$, and $R_D$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). …

50: 1.16 Distributions

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1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential operator (locally)

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, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

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

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

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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

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