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11: 15.4 Special Cases

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

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15.4.10 $$F(a,\frac{1}{2}+a;\frac{3}{2};-{\tan }^{2}z)={(\cos z)}^{2a}\frac{\sin \left((1-2a)z\right)}{(1-2a)\sin z}.$$

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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, $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E10

Encodings:

pMML, png, TeX

See also:

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

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15.4.12 $$F(-a,a;\frac{1}{2};{\sin }^{2}z)=\cos \left(2az\right).$$

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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, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E12

Encodings:

pMML, png, TeX

See also:

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

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15.4.14 $$F(a,1-a;\frac{1}{2};{\sin }^{2}z)=\frac{\cos \left((2a-1)z\right)}{\cos z}.$$

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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, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E14

Encodings:

pMML, png, TeX

See also:

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

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15.4.16 $$F(a,1-a;\frac{3}{2};{\sin }^{2}z)=\frac{\sin \left((2a-1)z\right)}{(2a-1)\sin z}.$$

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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, $\sin z$: sine function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E16

Encodings:

pMML, png, TeX

See also:

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

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12: 19.4 Derivatives and Differential Equations

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19.4.5 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-k_{}^{\prime }{}_{}{}^{2}F(\varphi ,k)}{kk_{}^{\prime }{}_{}{}^2}-\frac{k\sin \varphi \cos \varphi }{k_{}^{\prime }{}_{}{}^2\sqrt{1-k^2{\sin }^2\varphi }},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

… ►Let $D_k=\partial /\partial k$. Then ►

19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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13: 10.31 Power Series

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14: 1.10 Functions of a Complex Variable

… ►Let $F(z)$ be a multivalued function and $D$ be a domain. … ►Then $a$ is a branch point of $F(z)$. … ►Then the value of $F(z)$ at any other point is obtained by analytic continuation. … ►Then (1.10.12) has a solution $z=F(w)$, where … ►Then $F(z)$ is analytic in $D$. …

15: 31.10 Integral Equations and Representations

… ►for a suitable contour $C$. …The contour $C$ must be such that … ►Here ${\tilde{\kappa }}_m$ is a normalization constant and $C$ is the contour of Example 1. … ►for suitable contours $C_1$, $C_2$. …The contours $C_1$, $C_2$ must be chosen so that …

16: 4.43 Cubic Equations

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4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

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$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

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17: 19.7 Connection Formulas

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$F(\varphi ,k_1)=kF(\beta ,k),$

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$\mathrm{\Pi }(\varphi ,{\alpha }^2,k_1)=k\mathrm{\Pi }(\beta ,k^2{\alpha }^2,k),$ $k_1=1/k$, $\sin \beta =k_1\sin \varphi \le 1$.

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$$E(\varphi ,\mathrm{i}k)=(1/{\kappa }^{\prime })\left(E(\theta ,\kappa )-{\kappa }^2(\sin \theta \cos \theta ){(1-{\kappa }^2{\sin }^2\theta )}^{-1/2}\right),$$

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$\sin \theta ={\displaystyle \frac{\sqrt{1+k^2}\sin \varphi }{\sqrt{1+k^2{\sin }^2\varphi }}},$

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$$E(\mathrm{i}\varphi ,k)=\mathrm{i}\left(F(\psi ,k^{\prime })-E(\psi ,k^{\prime })+(\tan \psi )\sqrt{1-k_{}^{\prime }{}_{}{}^2{\sin }^2\psi }\right),$$

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18: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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$F_{\mathrm{\ell }}(\eta ,\rho )\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$

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$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )\sim (\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }}.$

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$F_{\mathrm{\ell }}(0,\rho )=\rho {𝗃}_{\mathrm{\ell }}\left(\rho \right),$

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$F_0(0,\rho )=\sin \rho ,$

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$F_{\mathrm{\ell }}(\eta ,\rho )\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$

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19: Bibliography V

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On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.

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External Links:

ISSN 0025-584X, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§28.6(i).

Encodings:

BibTeX

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Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

Encodings:

BibTeX

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20: 19.6 Special Cases

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§19.6(ii) $F(\varphi ,k)$

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

… ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. Let $c={\csc }^2\varphi \ne {\alpha }^2$ and $\mathrm{\Delta }=\sqrt{1-k^2{\sin }^2\varphi }$. … ►

§19.6(v) $R_C(x,y)$

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