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31: 36.4 Bifurcation Sets

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36.4.5 $$x=0.$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.6 $$27x^2=-8y^3.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.11 $$x+\mathrm{i}y=-z^2\exp \left(\frac{2}{3}\mathrm{i}\pi m\right),$$ $m=0,1,2$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $n$: integer and $x$: real parameter

Referenced by:

§36.7(iii)

Permalink:

http://dlmf.nist.gov/36.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.13 $$x=y=-\frac{1}{4}{z}^2.$$

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Symbols:

$y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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32: 16.22 Asymptotic Expansions

… ►For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).

33: 31.7 Relations to Other Functions

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31.7.1 $${}_2{}^{}F_1^{}(\alpha ,\beta ;\gamma ;z)=H\mathrm{\ell }(1,\alpha \beta ;\alpha ,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(0,0;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z)=H\mathrm{\ell }(a,a\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. Below are three such reductions with three and two parameters. … ►

31.7.2 $$H\mathrm{\ell }(2,\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta -2\gamma +1;z)={}_2{}^{}F_1^{}(\frac{1}{2}\alpha ,\frac{1}{2}\beta ;\gamma ;1-{(1-z)}^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

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31.7.3 $$H\mathrm{\ell }(4,\alpha \beta ;\alpha ,\beta ,\frac{1}{2},\frac{2}{3}(\alpha +\beta );z)={}_2{}^{}F_1^{}(\frac{1}{3}\alpha ,\frac{1}{3}\beta ;\frac{1}{2};1-{(1-z)}^2(1-\frac{1}{4}z)),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

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34: 31.10 Integral Equations and Representations

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31.10.2 $$\rho (t)=t^{\gamma -1}{(t-1)}^{\delta -1}{(t-a)}^{ϵ-1},$$

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Defines:

$\rho (t)$: weight function (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Permalink:

http://dlmf.nist.gov/31.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

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31.10.4 $${𝒟}_z=z(z-1)(z-a)({\partial }^2/{\partial z}^2)+\left(\gamma (z-1)(z-a)+\delta z(z-a)+ϵz(z-1)\right)(\partial /\partial z)+\alpha \beta z.$$

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Defines:

${𝒟}_z$: Heun’s operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.10(ii)

Permalink:

http://dlmf.nist.gov/31.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

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31.10.6 $$p(t)=t^{\gamma }{(t-1)}^{\delta }{(t-a)}^{ϵ}.$$

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Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Referenced by:

§31.10(ii)

Permalink:

http://dlmf.nist.gov/31.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

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31.10.10 $$𝒦(z,t)={(zt-a)}^{\frac{1}{2}-\delta -\sigma }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}-\delta -\sigma +\alpha ,\frac{1}{2}-\delta -\sigma +\beta }{\gamma };\frac{zt}{a}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}+\delta +\sigma ,-\frac{1}{2}+ϵ-\sigma }{\delta };\frac{a(z-1)(t-1)}{(a-1)(zt-a)}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel and $\sigma$: separation constant

Permalink:

http://dlmf.nist.gov/31.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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31.10.13 $$\rho (s,t)=(s-t){(st)}^{\gamma -1}{\left((1-s)(1-t)\right)}^{\delta -1}{\left((1-(s/a))(1-(t/a))\right)}^{ϵ-1},$$

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Defines:

$\rho (s,t)$: weight function (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Permalink:

http://dlmf.nist.gov/31.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10 and Ch.31

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35: 29.7 Asymptotic Expansions

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29.7.1 $$a_{\nu }^m\left(k^2\right)\sim p\kappa -{\tau }_0-{\tau }_1{\kappa }^{-1}-{\tau }_2{\kappa }^{-2}-\mathrm{\cdots },$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $\sim$: Poincaré asymptotic expansion, $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\kappa$ and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.3 $${\tau }_0=\frac{1}{2^3}(1+k^2)(1+p^2),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.4 $${\tau }_1=\frac{p}{2^6}({(1+k^2)}^2(p^2+3)-4k^2(p^2+5)).$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.5 $$b_{\nu }^{m+1}\left(k^2\right)-a_{\nu }^m\left(k^2\right)=O\left({\nu }^{m+\frac{3}{2}}{\left(\frac{1-k}{1+k}\right)}^{\nu }\right),$$ $\nu \to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Referenced by:

§29.7(i)

Permalink:

http://dlmf.nist.gov/29.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.6 $${\tau }_2=\frac{1}{2^{10}}(1+k^2){(1-k^2)}^2(5p^4+34p^2+9),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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36: 15.11 Riemann’s Differential Equation

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15.11.2 $$a_1+a_2+b_1+b_2+c_1+c_2=1.$$

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Symbols:

$a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.3

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Defines:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation

Symbols:

$z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.4

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.6

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $t$: mapping

Permalink:

http://dlmf.nist.gov/15.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

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15.11.8

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\lambda$: constant and $\mu$: constant

Permalink:

http://dlmf.nist.gov/15.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

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37: 36.3 Visualizations of Canonical Integrals

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38: 16.27 Software

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§16.27(ii) Real Argument and Parameters

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§16.27(iii) Complex Argument and/or Parameters

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39: 29.10 Lamé Functions with Imaginary Periods

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29.10.1 $$h=\nu (\nu +1)-h^{\prime },$$

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Symbols:

$h$: real parameter and $\nu$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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29.10.3 $$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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40: 36.1 Special Notation

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$l,m,n$	integers.
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$𝐱$	$\{x_1,x_2,\mathrm{\dots },x_K\}$, where $x_1,x_2,\mathrm{\dots },x_K$ are real parameters; also $x_1=x$, $x_2=y$, $x_3=z$ when $K\le 3$.
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