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1: 8.22 Mathematical Applications

… ►

§8.22(i) Terminant Function

►The so-called terminant function $F_p\left(z\right)$, defined by ►

8.22.1 $$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.22.E1

Encodings:

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See also:

Annotations for §8.22(i), §8.22 and Ch.8

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2: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.10 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{n-1}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}+{(-1)}^n\frac{2\pi }{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}{F}_{n+p}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $F_p\left(z\right)$: terminant function

Referenced by:

§2.11(iii), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E10

Encodings:

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See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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2.11.11 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{n+p-1}}{1+t}dt=\frac{\mathrm{\Gamma }\left(n+p\right)}{2\pi }\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.$$

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Defines:

$F_p\left(z\right)$: terminant function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.11.E11

Encodings:

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See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►In this context the $F$-functions are called terminants, a name introduced by Dingle (1973). …

3: 16.2 Definition and Analytic Properties

… ►Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$. … ►However, when one or more of the top parameters $a_j$ is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in $z$. Note that if $-m$ is the value of the numerically largest $a_j$ that is a nonpositive integer, then the identity …

4: 29.15 Fourier Series and Chebyshev Series

… ►When $\nu =2n$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.1) terminates: … ►When $\nu =2n+1$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.16) terminates: … ►When $\nu =2n+1$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.31) terminates: … ►When $\nu =2n+1$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.8) terminates: … ►When $\nu =2n+2$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.46) terminates: …

5: 9.7 Asymptotic Expansions

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9.7.20 $$R_n(z)={(-1)}^n\sum _{k=0}^{m-1}{(-1)}^k{u}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+R_{m,n}(z),$$

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Defines:

$R_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $u_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E20

Encodings:

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See also:

Annotations for §9.7(v), §9.7 and Ch.9

►

9.7.21 $$S_n(z)={(-1)}^{n-1}\sum _{k=0}^{m-1}{(-1)}^k{v}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+S_{m,n}(z),$$

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Defines:

$S_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $v_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

… ►

9.7.22 $$G_p\left(z\right)=\frac{{\mathrm{e}}^z}{2\pi }\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }(1-p,z).$$

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Defines:

$G_p\left(z\right)$: rescaled terminant function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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6: 34.6 Definition: $\mathit{9}j$ Symbol

… ►The $\mathit{9}j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

7: 10.17 Asymptotic Expansions for Large Argument

… ►If these expansions are terminated when $k=\mathrm{\ell }-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. … ►

10.17.16 $$G_p\left(z\right)=\frac{{\mathrm{e}}^z}{2\pi }\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }(1-p,z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $G_p\left(z\right)$: rescaled terminant function and $z$: complex variable

Referenced by:

§10.40(iv)

Permalink:

http://dlmf.nist.gov/10.17.E16

Encodings:

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See also:

Annotations for §10.17(v), §10.17 and Ch.10

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10.17.17 $$R_{\mathrm{\ell }}^{\pm }(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}{(\pm \mathrm{i})}^k\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(\mp 2\mathrm{i}z\right)+R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder

Permalink:

http://dlmf.nist.gov/10.17.E17

Encodings:

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See also:

Annotations for §10.17(v), §10.17 and Ch.10

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8: 4.6 Power Series

… ►If $a=0,1,2,\mathrm{\dots }$, then the series terminates and $z$ is unrestricted. …

9: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.13 $$R_{\mathrm{\ell }}(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(2z\right)+R_{m,\mathrm{\ell }}(\nu ,z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{m,\mathrm{\ell }}(\nu ,z)$: remainder

Permalink:

http://dlmf.nist.gov/10.40.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iv), §10.40 and Ch.10

►where $G_p\left(z\right)$ is given by (10.17.16). …

10: 16.4 Argument Unity

… ►See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … ►when $\mathrm{\Re }\left(a-2b-2c\right)>-2$, or when the series terminates with $a=-n$: … ►when $\mathrm{\Re }\left(2c-a-b\right)>-1$, or when the series terminates with $a=-n$. … ►when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. …