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1: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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2: 12.3 Graphics

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Figure 12.3.1: $U(a,x)$, $a$ = 0. …5, 5, 8. Magnify

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Figure 12.3.2: $V(a,x)$, $a$ = 0. …5, 5, 8. Magnify

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Figure 12.3.5: $U(-8,x)$, $\overline{U}(-8,x)$, $F(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

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Figure 12.3.6: $U^{\prime }(-8,x)$, ${\overline{U}}^{\prime }(-8,x)$, $G(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

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3: 15.4 Special Cases

… ►

$$F(a,b;a;z)={(1-z)}^{-b},$$

… ►where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when the third parameter is a nonpositive integer. … ►

15.4.31 $$F(a,\frac{1}{2}+a;\frac{3}{2}-2a;-\frac{1}{3})={\left(\frac{8}{9}\right)}^{-2a}\frac{\mathrm{\Gamma }\left(\frac{4}{3}\right)\mathrm{\Gamma }\left(\frac{3}{2}-2a\right)}{\mathrm{\Gamma }\left(\frac{3}{2}\right)\mathrm{\Gamma }\left(\frac{4}{3}-2a\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function and $a$: real or complex parameter

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.4.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

►

15.4.32 $$F(a,\frac{1}{2}+a;\frac{5}{6}+\frac{2}{3}a;\frac{1}{9})=\sqrt{\pi }{\left(\frac{3}{4}\right)}^a\frac{\mathrm{\Gamma }\left(\frac{5}{6}+\frac{2}{3}a\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{3}a\right)\mathrm{\Gamma }\left(\frac{5}{6}+\frac{1}{3}a\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $a$: real or complex parameter

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.4.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

… ►where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33) $a=-\frac{1}{3},-\frac{4}{3},-\frac{7}{3},\mathrm{\dots }$, and in (15.4.34) $a=0,-1,-2,\mathrm{\dots }$. …

4: 7.3 Graphics

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Figure 7.3.2: Dawson’s integral $F\left(x\right)$, $-3.5\le x\le 3.5$. Magnify

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Figure 7.3.3: Fresnel integrals $C\left(x\right)$ and $S\left(x\right)$, $0\le x\le 4$. Magnify

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. … Magnify

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5: 15.8 Transformations of Variable

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15.8.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={(1-z)}^{-a}𝐅(\genfrac{}{}{0pt}{}{a,c-b}{c};\frac{z}{z-1})={(1-z)}^{-b}𝐅(\genfrac{}{}{0pt}{}{c-a,b}{c};\frac{z}{z-1})={(1-z)}^{c-a-b}𝐅(\genfrac{}{}{0pt}{}{c-a,c-b}{c};z),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.10(ii), §15.13, §15.16, (15.4.27), §15.4(i), §15.4(iii), §15.8(i), §15.8(ii), §15.8(iii), §8.17(i)

Permalink:

http://dlmf.nist.gov/15.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

… ►Alternatively, if $b-a$ is a negative integer, then we interchange $a$ and $b$ in $𝐅(a,b;c;z)$. … ►

15.8.12 $$𝐅(a,b;a+b-m;z)={(1-z)}^{-m}𝐅(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z),$$ $\tilde{a}=a-m,\tilde{b}=b-m$.

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/15.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

… ►

15.8.13 $$F(\genfrac{}{}{0pt}{}{a,b}{2b};z)={\left(1-\frac{1}{2}z\right)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{b+\frac{1}{2}};{\left(\frac{z}{2-z}\right)}^2),$$ ,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Referenced by:

§15.8(iv)

Permalink:

http://dlmf.nist.gov/15.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

… ►

15.8.31 $$F(\genfrac{}{}{0pt}{}{3a,3a+\frac{1}{2}}{4a+\frac{2}{3}};z)={\left(1-\frac{9}{8}z\right)}^{-2a}F(\genfrac{}{}{0pt}{}{a,a+\frac{1}{2}}{2a+\frac{5}{6}};\frac{27z^2(z-1)}{{(9z-8)}^2}),$$ .

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§15.8(v)

Permalink:

http://dlmf.nist.gov/15.8.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(v), §15.8(v), §15.8 and Ch.15

…

6: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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$F_{\mathrm{\ell }}(\eta ,\rho )\sim {\displaystyle \frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(2\eta )}^{\mathrm{\ell }+1}}}{(2\eta \rho )}^{1/2}{I}_{2\mathrm{\ell }+1}\left({(8\eta \rho )}^{1/2}\right),$

… ►

$$F_0(\eta ,\rho )\sim {\mathrm{e}}^{-\pi \eta }{(\pi \rho )}^{1/2}{I}_1\left({(8\eta \rho )}^{1/2}\right),$$

… ►

$$F_0^{\prime }(\eta ,\rho )\sim {\mathrm{e}}^{-\pi \eta }{(2\pi \eta )}^{1/2}{I}_0\left({(8\eta \rho )}^{1/2}\right),$$

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$$F_0(\eta ,\rho )={(\pi \rho )}^{1/2}{J}_1\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{-1/4}\right),$$

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$$F_0^{\prime }(\eta ,\rho )={(-2\pi \eta )}^{1/2}{J}_0\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{1/4}\right),$$

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7: 25.20 Approximations

… ►

Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

… ►

Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

8: 33.20 Expansions for Small $|ϵ|$

… ►where ►

33.20.4 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{J}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

… ►The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and … ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and …The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). …

9: 16.4 Argument Unity

… ►The function ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ is well-poised if … ►The function ${}_{q+1}{}^{}F_q^{}$ with argument unity and general values of the parameters is discussed in Bühring (1992). … ►For generalizations involving ${}_{r+3}{}^{}F_{r+2}^{}$ functions see Kim et al. (2013). … ►Transformations for both balanced ${}_4{}^{}F_3^{}\left(1\right)$ and very well-poised ${}_7{}^{}F_6^{}\left(1\right)$ are included in Bailey (1964, pp. 56–63). A similar theory is available for very well-poised ${}_9{}^{}F_8^{}\left(1\right)$’s which are 2-balanced. …

10: 18.5 Explicit Representations

… ►In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_n$. … ►In this equation $w(x)$ is as in Table 18.3.1, (reproduced in Table 18.5.1), and $F(x)$, ${\kappa }_n$ are as in Table 18.5.1. … ►For the definitions of ${}_2{}^{}F_1^{}$, ${}_1{}^{}F_1^{}$, and ${}_2{}^{}F_0^{}$ see §16.2. … ►The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of $P_n^{(\alpha ,\beta )}\left(x\right)$ when the conditions $\alpha >-1$ and $\beta >-1$ are not satisfied. …Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. …