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11: 31.14 General Fuchsian Equation

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31.14.1

{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},

\sum_{j=1}^{N}q_{j}=0

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14 and Ch.31

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31.14.3

w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),

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Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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31.14.4

\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{% \gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,

\sum_{j=1}^{N}\tilde{q}_{j}=0

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Defines:: $W(z)$ : solution (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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12: 15.1 Special Notation

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$x$	real variable.
$z=x+\mathrm{i}y$	complex variable.
$a, b, c$	real or complex parameters.
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15.1.1

{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for 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/15.1.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

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15.1.2

\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s 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/15.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

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13: 12.8 Recurrence Relations and Derivatives

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12.8.1

zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,

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Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.4
Referenced by:: §12.8(i)
Permalink:: http://dlmf.nist.gov/12.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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12.8.2

U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,

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Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.1
Permalink:: http://dlmf.nist.gov/12.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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12.8.3

U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,

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Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.2
Permalink:: http://dlmf.nist.gov/12.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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12.8.4

2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0.

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Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.3
Referenced by:: §12.8(i)
Permalink:: http://dlmf.nist.gov/12.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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12.8.5

zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-1,z\right)=0,

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Symbols:: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.6.8
Permalink:: http://dlmf.nist.gov/12.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §12.8(i), §12.8 and Ch.12

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14: 25.1 Special Notation

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$k, m, n$	nonnegative integers.
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$a$	real or complex parameter.
$s=\sigma+it$	complex variable.
$z=x+iy$	complex variable.
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15: 31.4 Solutions Analytic at Two Singularities: Heun Functions

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31.4.1

(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),

m=0,1,2,\dots

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Symbols:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.6
Permalink:: http://dlmf.nist.gov/31.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.4 and Ch.31

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31.4.2

q=\cfrac{a\gamma P_{1}}{Q_{1}+q-\cfrac{R_{1}P_{2}}{Q_{2}+q-\cfrac{R_{2}P_{3}}{% Q_{3}+q-\cdots}}},

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Symbols:: $\gamma$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Referenced by:: §31.18
Permalink:: http://dlmf.nist.gov/31.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.4 and Ch.31

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31.4.3

(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),

m=0,1,2,\dots

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Symbols:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.4, §31.6
Permalink:: http://dlmf.nist.gov/31.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.4 and Ch.31

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16: 5.1 Special Notation

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$j, m, n$	nonnegative integers.
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$z=x+iy$	complex variable.
$a, b, q, s, w$	real or complex variables with $\|q\|<1$ .
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17: 5.24 Software

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§5.24(iv) $\Gamma\left(z\right)$ , $\psi\left(z\right)$ , $\psi^{(n)}\left(z\right)$ , $z\in\mathbb{C}$

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§5.24(vi) $\mathrm{B}\left(a,b\right)$ , $a,b\in\mathbb{C}$

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18: 20.16 Software

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§20.16(iii) Complex Argument and/or Parameter

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19: 21.8 Abelian Functions

… ►An Abelian function is a

2g

-fold periodic, meromorphic function of

g

complex variables. In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

20: 22.22 Software

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§22.22(iii) Complex Argument

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