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11: 31.9 Orthogonality

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

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See also:

Annotations for §31.9(i), §31.9 and Ch.31

… ►The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). … ►

§31.9(ii) Double Orthogonality

… ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

12: 8.13 Zeros

… ►As $x$ increases the positive zeros coalesce to form a double zero at ($a_n^{\ast },x_n^{\ast }$). The values of the first six double zeros are given to 5D in Table 8.13.1. … ►

Table 8.13.1: Double zeros $(a_n^{\ast },x_n^{\ast })$ of ${\gamma }^{\ast }(a,x)$.

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$n$	$a_n^{\ast }$		$x_n^{\ast }$
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13: 10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). …

14: Mathematical Introduction

… ►In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). … ► ►►►►►

$ℂ$	complex plane (excluding infinity).
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$⟹$	implies.
$⟺$	is equivalent to.
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$n!!$	double factorial: $2\cdot 4\cdot 6\mathrm{\cdots }n$ if $n=2,4,6,\mathrm{\dots }$; $1\cdot 3\cdot 5\mathrm{\cdots }n$ if $n=1,3,5,\mathrm{\dots }$; 1 if $n=0,-1$.
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15: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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33.5.6 $$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial, $!$: factorial (as in $n!$) and $\mathrm{\ell }$: nonnegative integer

Permalink:

http://dlmf.nist.gov/33.5.E6

Encodings:

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Annotations for §33.5(ii), §33.5 and Ch.33

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33.5.9 $$C_{\mathrm{\ell }}\left(\eta \right)\sim \frac{{\mathrm{e}}^{-\pi \eta /2}}{(2\mathrm{\ell }+1)!!}\sim {\mathrm{e}}^{-\pi \eta /2}\frac{{\mathrm{e}}^{\mathrm{\ell }}}{\sqrt{2}{(2\mathrm{\ell })}^{\mathrm{\ell }+1}}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.23(iv), §33.5(iv)

Permalink:

http://dlmf.nist.gov/33.5.E9

Encodings:

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Annotations for §33.5(iv), §33.5 and Ch.33

16: 1.5 Calculus of Two or More Variables

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§1.5(v) Multiple Integrals

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Double Integrals

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Infinite Double Integrals

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Change of Variables

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17: Guide to Searching the DLMF

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any double-quoted sequence of textual words and numbers.

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Table 3: A sample of recognized symbols

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Symbols	Comments
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->, <-, <->, =>, <==, <=>	For arrows $\to$, $\leftarrow$, $\leftrightarrow$, $\Rightarrow$, $\Leftarrow$ and $\iff$
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18: Bibliography U

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Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.

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External Links:

ISSN 0305-0041, Document, MathReview (F. I. Ibragimov), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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19: 1.9 Calculus of a Complex Variable

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§1.9(vii) Inversion of Limits

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Double Sequences and Series

… ►A double series is the limit of the double sequence …If the limit exists, then the double series is convergent; otherwise it is divergent. … …

20: 21.5 Modular Transformations

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21.5.5

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $𝛀$: a Riemann matrix and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E5

Encodings:

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See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.6

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $g$: positive integer, $𝛀$: a Riemann matrix and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E6

Encodings:

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See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.7

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $g$: positive integer, $𝛀$: a Riemann matrix, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Referenced by:

(21.5.8), Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/21.5.E7

Encodings:

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See also:

Annotations for §21.5(i), §21.5 and Ch.21

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$\Rightarrow$  

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