%E8%8F%B2%E5%BE%8B%E5%AE%BE%E7%BD%91%E4%B8%8A%E8%B5%8C%E5%9C%BA,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E8%B5%8C%E5%9C%BA%E5%9C%B0%E5%9D%80,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E9%A9%AC%E5%B0%BC%E6%8B%89%E8%B5%8C%E5%9C%BA,%20%E3%80%90%E5%8D%9A%E5%BD%A9%E7%BD%91%E5%9D%80%E2%88%B633kk88.com%E3%80%91%E6%AD%A3%E8%A7%84%E5%8D%9A%E5%BD%A9%E5%B9%B3%E5%8F%B0,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E5%8D%9A%E5%BD%A9%E5%85%AC%E5%8F%B8,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E5%8D%9A%E5%BD%A9%E7%BD%91%E7%AB%99,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E7%BA%BF%E4%B8%8A%E8%B5%8C%E5%9C%BA%E3%80%90%E8%B5%8C%E5%9C%BA%E5%9C%B0%E5%9D%80%E2%88%B633kk88.com%E3%80%91%E7%BD%91%E5%9D%80ZCg0B0fCAg00fnE

SearchAdvancedHelp

Did you mean %E8%8F%B2%E5%BE%8B%E5%AE%BE%E7%BD%91%E4%B8%8A%E8%B5%8C%E5%9C%BA,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E8%B5%8C%E5%9C%BA%E5%9C%B0%E5%9D%80,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E9%A9%AC%E5%B0%BC%E6%8B%89%E8%B5%8C%E5%9C%BA,%20%E3%80%90%E5%8D%9A%E5%BD%A9%E7%BD%91%E5%9D%80%E2%88%gcn.com%E3%80%91%E6%AD%A3%E8%A7%84%E5%8D%9A%E5%BD%A9%E5%B9%B3%E5%8F%B0,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E5%8D%9A%E5%BD%A9%E5%85%AC%E5%8F%B8,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E5%8D%9A%E5%BD%A9%E7%BD%91%E7%AB%99,%E8%8F%B2%E5%BE%8B%E5%AE%BE%E7%BA%BF%E4%B8%8A%E8%B5%8C%E5%9C%BA%E3%80%90%E8%B5%8C%E5%9C%BA%E5%9C%B0%E5%9D%80%E2%88%gcn.com%E3%80%91%E7%BD%91%E5%9D%80ZCg0B0fCAg00fnE ?

(0.085 seconds)

1—10 of 769 matching pages

1: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

2: 20 Theta Functions

Chapter 20 Theta Functions

…

3: 12.3 Graphics

… ►

►►

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

►

►►

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

… ►

►►

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

►

►►

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

…

4: 36 Integrals with Coalescing Saddles

…

5: 25.20 Approximations

… ►

Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

►

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 ${10}^{-4},{10}^{-8},{10}^{-12}$.

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

ⓘ

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

ⓘ

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) $a=0,-1,-2,\mathrm{\dots }$. …

7: 7.3 Graphics

… ►

►►

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

►

►►

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

►

►►

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

…

8: 15.8 Transformations of Variable

… ►

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

ⓘ

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

ⓘ

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

ⓘ

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

ⓘ

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

…

9: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

… ►

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

… ►

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

… ►

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

… ►

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

… ►

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

…

10: 8.17 Incomplete Beta Functions

… ►where, as in §5.12, $\mathrm{B}(a,b)$ denotes the beta function: … ►For a historical profile of ${\mathrm{B}}_x(a,b)$ see Dutka (1981). … ►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). … ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. … ►

8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

…