of a complex variable

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1: 12.4 Power-Series Expansions

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12.4.1

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

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Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $u_{1}(a,z)$ : solution and $u_{2}(a,z)$ : solution
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

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12.4.2

V\left(a,z\right)=V\left(a,0\right)u_{1}(a,z)+V'\left(a,0\right)u_{2}(a,z),

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Symbols:: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $u_{1}(a,z)$ : solution and $u_{2}(a,z)$ : solution
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

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12.4.3

u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+% \tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),

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Defines:: $u_{1}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.1 (modification of)
Referenced by:: §12.14(v), §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

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12.4.5

u_{1}(a,z)=e^{\tfrac{1}{4}z^{2}}\left(1+(a-\tfrac{1}{2})\frac{z^{2}}{2!}+(a-% \tfrac{1}{2})(a-\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $a$ : real or complex parameter and $u_{1}(a,z)$ : solution
A&S Ref:: 19.2.2 (modification of)
Permalink:: http://dlmf.nist.gov/12.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

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12.4.6

u_{2}(a,z)=e^{\tfrac{1}{4}z^{2}}\left(z+(a-\tfrac{3}{2})\frac{z^{3}}{3!}+(a-% \tfrac{3}{2})(a-\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $a$ : real or complex parameter and $u_{2}(a,z)$ : solution
A&S Ref:: 19.2.4 (modification of)
Referenced by:: §12.14(v), §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §12.4 and Ch.12

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2: 15.7 Continued Fractions

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15.7.1

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},

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Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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15.7.3

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},

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Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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15.7.5

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},

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Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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3: 15.5 Derivatives and Contiguous Functions

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15.5.11

(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z% -1)F\left(a+1,b;c;z\right)=0,

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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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.4(i), §15.5(ii)
Permalink:: http://dlmf.nist.gov/15.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

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15.5.12

(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0,

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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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

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15.5.13

(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F\left(a,b-1;c% ;z\right)=0,

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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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

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15.5.14

c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-% a)(c-b)zF\left(a,b;c+1;z\right)=0,

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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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

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15.5.15

(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-(c-1)F\left(a,b;c-1;z% \right)=0,

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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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.5.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(ii), §15.5 and Ch.15

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4: 5.1 Special Notation

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$j, m, n$	nonnegative integers.
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$a, b, q, s, w$	real or complex variables with $\|q\|<1$ .
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5: 8.8 Recurrence Relations and Derivatives

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8.8.3

w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.

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Symbols:: $z$ : complex variable, $a$ : parameter and $w(a,z)$ : function
Permalink:: http://dlmf.nist.gov/8.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §8.8 and Ch.8

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8.8.11

P\left(a+n,z\right)=P\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)},

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/8.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §8.8 and Ch.8

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8.8.12

Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)}.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/8.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §8.8 and Ch.8

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8.8.15

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\gamma\left(a+n,z\right),

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/8.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §8.8 and Ch.8

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8.8.16

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\Gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\Gamma\left(a+n,z\right),

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $n$ : nonnegative integer
A&S Ref:: 6.5.26
Referenced by:: §8.19(v)
Permalink:: http://dlmf.nist.gov/8.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §8.8 and Ch.8

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6: 15.1 Special Notation

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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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7: 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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8: 31.17 Physical Applications

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31.17.2

\frac{x_{s}^{2}}{z_{k}}+\frac{x_{t}^{2}}{z_{k}-1}+\frac{x_{u}^{2}}{z_{k}-a}=0,

k=1,2

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Symbols:: $z$ : complex variable, $a$ : complex parameter and $x_{s}$ , $x_{t}$ , $x_{u}$ : coordinates
Permalink:: http://dlmf.nist.gov/31.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.17(i), §31.17 and Ch.31

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31.17.4

\Psi(\mathbf{x})=(z_{1}z_{2})^{-s-\frac{1}{4}}((z_{1}-1)(z_{2}-1))^{-t-\frac{1% }{4}}\*((z_{1}-a)(z_{2}-a))^{-u-\frac{1}{4}}w(z_{1})w(z_{2}),

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Symbols:: $z$ : complex variable, $a$ : complex parameter and $\Psi(\mathbf{x})$ : common eigenfunction
Permalink:: http://dlmf.nist.gov/31.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.17(i), §31.17 and Ch.31

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9: 15.11 Riemann’s Differential Equation

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15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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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15.11.7

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=\left(\frac{z-\alpha}{z-\gamma}\right)^{a_{1}}% \left(\frac{z-\beta}{z-\gamma}\right)^{b_{1}}P\begin{Bmatrix}0&1&\infty&\\ 0&0&a_{1}+b_{1}+c_{1}&\dfrac{(z-\alpha)(\beta-\gamma)}{(z-\gamma)(\beta-\alpha% )}\\ a_{2}-a_{1}&b_{2}-b_{1}&a_{1}+b_{1}+c_{2}&\end{Bmatrix}.

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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 and $\gamma$ : point
Permalink:: http://dlmf.nist.gov/15.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

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15.11.8

z^{\lambda}(1-z)^{\mu}P\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}0&1&\infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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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10: 15.10 Hypergeometric Differential Equation

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15.10.1

z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=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, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)
Permalink:: http://dlmf.nist.gov/15.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10 and Ch.15

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15.10.3

\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(1-c)z^{-c}(1-z)^{c-a-b-1}.

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Symbols:: $\mathscr{W}$ : Wronskian, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions
Permalink:: http://dlmf.nist.gov/15.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.5

\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(a+b-c)z^{-c}(1-z)^{c-a-b-1}.

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Symbols:: $\mathscr{W}$ : Wronskian, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions
Permalink:: http://dlmf.nist.gov/15.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.7

\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(a-b)z^{-c}(z-1)^{c-a-b-1}.

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Symbols:: $\mathscr{W}$ : Wronskian, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions
Permalink:: http://dlmf.nist.gov/15.10.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.21

w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a% \right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a% +b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{3}(z)$ : Kummer’s solution $w_{3}$ and $w_{4}(z)$ : Kummer’s solution $w_{4}$
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.10.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

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