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21: 5.19 Mathematical Applications

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5.19.2 $$a_k=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}=\left(\frac{1}{k+1}-\frac{1}{k+\frac{1}{2}}\right)-2\left(\frac{1}{k+1}-\frac{1}{k+\frac{2}{3}}\right).$$

ⓘ

Symbols:

$k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

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5.19.3 $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\ln 3-2\ln 2-\frac{1}{3}\pi \sqrt{3}.$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/5.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

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$V={\displaystyle \frac{{\pi }^{\frac{1}{2}n}{r}^n}{\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}},$

►

$S={\displaystyle \frac{2{\pi }^{\frac{1}{2}n}{r}^{n-1}}{\mathrm{\Gamma }\left(\frac{1}{2}n\right)}}={\displaystyle \frac{n}{r}}V.$

22: 13.23 Integrals

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13.23.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)}{{\left(z+\frac{1}{2}\right)}^{\mu +\nu +\frac{1}{2}}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}+\mu +\nu }{1+2\mu };\frac{1}{z+\frac{1}{2}}),$$ $\mathrm{\Re }\mu +\nu +\frac{1}{2}>0$, $\mathrm{\Re }z>\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

►

13.23.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(2\mu +1\right){\left(z+\frac{1}{2}\right)}^{-\kappa -\mu -\frac{1}{2}}{\left(z-\frac{1}{2}\right)}^{\kappa -\mu -\frac{1}{2}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }z>\frac{1}{2}$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.9 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\mu -\frac{1}{2}(\nu +1)}{M}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\kappa +\nu \right)}{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\kappa -\mu -\frac{3}{2})}{M}_{\frac{1}{2}(\kappa +3\mu -\nu +\frac{1}{2}),\frac{1}{2}(\kappa -\mu +\nu -\frac{1}{2})}\left(x\right),$$ $x>0$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iii), §13.23 and Ch.13

… ►Let $f(x)$ be absolutely integrable on the interval $[r,R]$ for all positive , $f(x)=O\left(x^{{\rho }_0}\right)$ as $x\to 0+$, and $f(x)=O\left({\mathrm{e}}^{-{\rho }_{1}x}\right)$ as $x\to +\mathrm{\infty }$, where ${\rho }_1>\frac{1}{2}$. Then for $\mu$ in the half-plane $\mathrm{\Re }\mu \ge {\mu }_1>\max (-{\rho }_0,\mathrm{\Re }\kappa -\frac{1}{2})$ …

23: 24.6 Explicit Formulas

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24.6.2 $$B_n=\frac{1}{n+1}\sum _{k=1}^n\sum _{j=1}^k{(-1)}^j{j}^n\left(\genfrac{}{}{0.0pt}{}{n+1}{k-j}\right)/\left(\genfrac{}{}{0.0pt}{}{n}{k}\right),$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

►

24.6.7 $$B_n\left(x\right)=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(x+j)}^n,$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.12 $$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

24: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

► ►►►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
…
2	$\begin{array}{c}{\left(\sin \frac{1}{2}x\right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}x\right)}^{\beta +\frac{1}{2}}\\ \times P_n^{(\alpha ,\beta )}\left(\cos x\right)\end{array}$	$1$	$0$	${\displaystyle \frac{\frac{1}{4}-{\alpha }^2}{4{\sin }^2\frac{1}{2}x}}+{\displaystyle \frac{\frac{1}{4}-{\beta }^2}{4{\cos }^2\frac{1}{2}x}}$	${\left(n+\frac{1}{2}(\alpha +\beta +1)\right)}^2$
3	${(\sin x)}^{\alpha +\frac{1}{2}}{P}_n^{(\alpha ,\alpha )}\left(\cos x\right)$	$1$	$0$	$(\frac{1}{4}-{\alpha }^2)/{\sin }^{2}x$	${(n+\alpha +\frac{1}{2})}^2$
…
9	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(x^2\right)$	$1$	$0$	$-x^2+(\frac{1}{4}-{\alpha }^2)x^{-2}$	$4n+2\alpha +2$
10	${\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}\alpha }{L}_n^{(\alpha )}\left(x\right)$	$x$	$1$	$-\frac{1}{4}x-\frac{1}{4}{\alpha }^2{x}^{-1}$	$n+\frac{1}{2}(\alpha +1)$
…
13	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{H}_n\left(x\right)$	$1$	$0$	$-x^2$	$2n+1$
…

►

…

25: 13.16 Integral Representations

… ►If $\frac{1}{2}+\mu -\kappa \ne 0,-1,-2,\mathrm{\dots }$, then …where the contour of integration separates the poles of $\mathrm{\Gamma }\left(t-\kappa \right)$ from those of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu -t\right)$. ►If $\frac{1}{2}\pm \mu -\kappa \ne 0,-1,-2,\mathrm{\dots }$, then …where the contour of integration separates the poles of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)$ from those of $\mathrm{\Gamma }\left(-\kappa -t\right)$. …where the contour of integration passes all the poles of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)$ on the right-hand side.

26: 36.6 Scaling Relations

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$\text{umbilics: }{\beta }^{(\mathrm{U})}={\displaystyle \frac{1}{3}}.$

… ►

$\text{umbilics: }{\gamma }_x^{(\mathrm{U})}=\frac{2}{3},$

… ►

Table 36.6.1: Special cases of scaling exponents for cuspoids.

► ►►►►►

singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
fold	1	$\frac{1}{6}$	$\frac{2}{3}$	$-$	$-$	$\frac{2}{3}$
cusp	2	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{1}{2}$	$-$	$\frac{5}{4}$
swallowtail	3	$\frac{3}{10}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{9}{5}$

►

…

27: 11.4 Basic Properties

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11.4.5 $${𝐇}_{\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}(1-\cos z),$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $z$: complex variable

A&S Ref:

12.1.16

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

►

11.4.6 $${𝐇}_{-\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sin z,$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

… ►and $|{\nu }_0+\frac{3}{2}|$ is the smallest of the numbers $|\nu +\frac{3}{2}|$, $|\nu +\frac{5}{2}|$, $|\nu +\frac{9}{2}|,\mathrm{\dots }$. … ►

11.4.20 $${𝐇}_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu +\frac{1}{2}}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(k+\nu +\frac{1}{2})}{J}_{k+\frac{1}{2}}\left(z\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iv), §11.4 and Ch.11

… ►

11.4.22 $${𝐇}_1\left(z\right)=\frac{2}{\pi }(1-J_0\left(z\right))+\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{J_{2k}\left(z\right)}{4k^2-1}=4\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+\frac{1}{2}}\left(\frac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\frac{1}{2}z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: nonnegative integer

A&S Ref:

12.1.20 (First part)

Permalink:

http://dlmf.nist.gov/11.4.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(iv), §11.4 and Ch.11

…

28: 10.65 Power Series

… ►

${\mathrm{ber}}_{\nu }x={(\frac{1}{2}x)}^{\nu }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(\frac{3}{4}\nu \pi +\frac{1}{2}k\pi \right)}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}}{(\frac{1}{4}{x}^2)}^k,$

… ►

$\mathrm{bei}x=\frac{1}{4}{x}^2-{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^3}{{(3!)}^2}}+{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^5}{{(5!)}^2}}-\mathrm{\cdots }.$

… ►

10.65.3 $${\ker }_{n}x=\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{ber}}_{n}x+\frac{1}{4}\pi {\mathrm{bei}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k,$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

►

10.65.4 $${\mathrm{kei}}_{n}x=-\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{bei}}_{n}x-\frac{1}{4}\pi {\mathrm{ber}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k.$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

►

$$\ker x=-\ln \left(\frac{1}{2}x\right)\mathrm{ber}x+\frac{1}{4}\pi \mathrm{bei}x+\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(2k+1\right)}{{((2k)!)}^2}{(\frac{1}{4}{x}^2)}^{2k},$$

…

29: 10.70 Zeros

… ►

10.70.1 $$\frac{\mu -1}{16t}+\frac{\mu -1}{32t^2}+\frac{(\mu -1)(5\mu +19)}{1536t^3}+\frac{3{(\mu -1)}^2}{512t^4}+\mathrm{\cdots }.$$

ⓘ

Defines:

$f(t)$: function (locally)

A&S Ref:

9.10.35

Referenced by:

§10.70

Permalink:

http://dlmf.nist.gov/10.70.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.70 and Ch.10

… ►

$$\text{zeros of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

►

$$\text{zeros of }{\mathrm{bei}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu +\frac{1}{8})\pi$,

►

$$\text{zeros of }{\ker }_{\nu }x\sim \sqrt{2}(t+f(-t)),$$ $t=(m-\frac{1}{2}\nu -\frac{5}{8})\pi$,

►

$$\text{zeros of }{\mathrm{kei}}_{\nu }x\sim \sqrt{2}(t+f(-t)),$$ $t=(m-\frac{1}{2}\nu -\frac{1}{8})\pi$.

…

30: 15.8 Transformations of Variable

… ►

15.8.26 $$F(\genfrac{}{}{0pt}{}{a,1-a}{c};z)={(1-z)}^{c-1}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}(c-a+1)\right)\mathrm{\Gamma }\left(\frac{1}{2}c+\frac{1}{2}a\right)}F(\genfrac{}{}{0pt}{}{\frac{1}{2}c-\frac{1}{2}a,\frac{1}{2}c+\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}};{(1-2z)}^2)+(1-2z){(1-z)}^{c-1}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(-\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}c-\frac{1}{2}a\right)\mathrm{\Gamma }\left(\frac{1}{2}(c+a-1)\right)}F(\genfrac{}{}{0pt}{}{\frac{1}{2}c-\frac{1}{2}a+\frac{1}{2},\frac{1}{2}c+\frac{1}{2}a}{\frac{3}{2}};{(1-2z)}^2),$$ , .

ⓘ

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, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.8.E26

Encodings:

pMML, png, TeX

See also:

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

… ► $b=\frac{1}{3}a+\frac{1}{3}$, $c=2b=a-b+1$ in Groups 1 and 2. … ►

15.8.29 $$F(\genfrac{}{}{0pt}{}{a,\frac{1}{3}a+\frac{1}{3}}{\frac{2}{3}a+\frac{2}{3}};z)={\left(1+\sqrt{z}\right)}^{-2a}F(\genfrac{}{}{0pt}{}{a,\frac{2}{3}a+\frac{1}{6}}{\frac{4}{3}a+\frac{1}{3}};\frac{4\sqrt{z}}{{(1+\sqrt{z})}^2}).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/15.8.E29

Encodings:

pMML, png, TeX

See also:

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

… ►

15.8.30 $${\left(1-\frac{1}{2}z\right)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{\frac{1}{3}a+\frac{5}{6}};{\left(\frac{z}{2-z}\right)}^2)=F(\genfrac{}{}{0pt}{}{a,\frac{1}{3}a+\frac{1}{3}}{\frac{2}{3}a+\frac{2}{3}};z)={(1+z)}^{-a}F(\genfrac{}{}{0pt}{}{\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}}{\frac{2}{3}a+\frac{2}{3}};\frac{4z}{{(1+z)}^2}),$$

ⓘ

Symbols:

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

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.8.E30

Encodings:

pMML, png, TeX

See also:

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

… ►provided that $z$ lies in the intersection of the open disks , or equivalently, . …