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11: 6.18 Methods of Computation

… ►Quadrature of the integral representations is another effective method. For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. …

12: 15.10 Hypergeometric Differential Equation

… ►

$f_1(z)=F({\displaystyle \genfrac{}{}{0pt}{}{a,b}{c}};z),$

… ►

$f_1(z)=F({\displaystyle \genfrac{}{}{0pt}{}{a,b}{a+b+1-c}};1-z),$

… ►(b) If $c$ equals $n=1,2,3,\mathrm{\dots }$, and $a\ne 1,2,\mathrm{\dots },n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by $F(a,b;n;z)$ and … ►

15.10.11 $$w_1(z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z)={(1-z)}^{c-a-b}F(\genfrac{}{}{0pt}{}{c-a,c-b}{c};z)={(1-z)}^{-a}F(\genfrac{}{}{0pt}{}{a,c-b}{c};\frac{z}{z-1})={(1-z)}^{-b}F(\genfrac{}{}{0pt}{}{c-a,b}{c};\frac{z}{z-1}).$$

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Defines:

$w_1(z)$: Kummer’s solution $w_1$ (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/15.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

13: 18.5 Explicit Representations

… ►For the definitions of ${}_2{}^{}F_1^{}$, ${}_1{}^{}F_1^{}$, and ${}_2{}^{}F_0^{}$ see §16.2. … ►

18.5.9 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.46

Referenced by:

§18.5(iii), §18.6(i)

Permalink:

http://dlmf.nist.gov/18.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11_2 $$T_n\left(x\right)={}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n}{\frac{1}{2}};\frac{1-x}{2}),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.5.E11_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.11_4 $$U_n\left(x\right)=\left(n+1\right){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+2}{\frac{3}{2}};\frac{1-x}{2}).$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.5(iii), Erratum (V1.2.0) Section 18.5

Permalink:

http://dlmf.nist.gov/18.5.E11_4

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

$T_5\left(x\right)=16x^5-20x^3+5x,$

…

14: 8.17 Incomplete Beta Functions

… ►

8.17.7 $${\mathrm{B}}_x(a,b)=\frac{x^a}{a}F(a,1-b;a+1;x),$$

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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, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.8

Permalink:

http://dlmf.nist.gov/8.17.E7

Encodings:

pMML, png, TeX

See also:

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

►

8.17.8 $${\mathrm{B}}_x(a,b)=\frac{x^a{(1-x)}^b}{a}F(a+b,1;a+1;x),$$

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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, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E8

Encodings:

pMML, png, TeX

See also:

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

►

8.17.9 $${\mathrm{B}}_x(a,b)=\frac{x^a{(1-x)}^{b-1}}{a}F(\genfrac{}{}{0pt}{}{1,1-b}{a+1};\frac{x}{x-1}).$$

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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, ${\mathrm{B}}_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E9

Encodings:

pMML, png, TeX

See also:

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

►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). … ►

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; .

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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

…

15: 15.3 Graphics

… ►

►►

Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

►

►►

Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

►

►►

Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

►

►►

Figure 15.3.4: $F(5,10;1;x),-1\le x\le 0.022$. Magnify

… ►

►

►

Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

…

16: Bibliography C

… ►

A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

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Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

Encodings:

BibTeX

… ►

Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(14.13.2), §14.3(iii), §14.3(iii).

Encodings:

BibTeX

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:

ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

Encodings:

BibTeX

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:

ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

Encodings:

BibTeX

…

17: 16.12 Products

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16.12.1 $${}_0{}^{}F_1^{}(-;a;z){}_0{}^{}F_1^{}(-;b;z)={}_2{}^{}F_3^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1};4z).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

(10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8

Permalink:

http://dlmf.nist.gov/16.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

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16.12.2 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b+\frac{1}{2}};z)\right)}^2={}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b};z).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)

Permalink:

http://dlmf.nist.gov/16.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

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16.12.3 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};z)\right)}^2=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(2a\right)}_k{\left(2b\right)}_k{\left(c-\frac{1}{2}\right)}_k}{{\left(c\right)}_k{\left(2c-1\right)}_{k}k!}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1)z^k,$$ .

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12

Permalink:

http://dlmf.nist.gov/16.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

…

18: 19.36 Methods of Computation

… ►The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta =1$ (leading ultimately to a hyperbolic case of $R_C$) or a descending Gauss transformation if $\theta =-1$ (leading to a circular case of $R_C$). … ►Descending Gauss transformations of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function $\mathrm{el2}(x,k_c,a,b)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …

19: Bibliography L

… ►

Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.

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External Links:

ISSN 0022-2526, Document, FullText, MathReview (M. S. Mahadeva Naika), ZentralBlatt

Referenced by:

§15.12(ii), §15.12(ii).

Encodings:

BibTeX

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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External Links:

ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

Encodings:

BibTeX

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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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External Links:

FullText, MathReview (J. Bryant), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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20: 15.4 Special Cases

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15.4.1 $$F(1,1;2;z)=-z^{-1}\ln \left(1-z\right),$$

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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, $\ln z$: principal branch of logarithm function and $z$: complex variable

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E1

Encodings:

pMML, png, TeX

See also:

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

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15.4.2 $$F(\frac{1}{2},1;\frac{3}{2};z^2)=\frac{1}{2z}\ln \left(\frac{1+z}{1-z}\right),$$

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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, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E2

Encodings:

pMML, png, TeX

See also:

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

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15.4.3 $$F(\frac{1}{2},1;\frac{3}{2};-z^2)=z^{-1}\arctan z,$$

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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, $\arctan z$: arctangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E3

Encodings:

pMML, png, TeX

See also:

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

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

►

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

…