phase shift (or phase)

(0.002 seconds)

31—40 of 42 matching pages

31: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.3

J_{\nu}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\*\left(% \cos\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{2k}(\nu)}{z^{2k}}-\sin\omega\sum% _{k=0}^{\infty}(-1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $a_{k}(\nu)$ : polynomial coefficient and $\omega$
A&S Ref:: 9.2.5
Referenced by:: §10.17(i), §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.49(i), §10.61(v), §10.7(ii)
Permalink:: http://dlmf.nist.gov/10.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.17(i), §10.17 and Ch.10

… ►

10.17.7

z^{\frac{1}{2}}=\exp\left(\tfrac{1}{2}\ln|z|+\tfrac{1}{2}i\operatorname{ph}z% \right).

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §10.17(i), §10.17 and Ch.10

►Corresponding expansions for other ranges of

\operatorname{ph}z

can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4). … ►

10.17.15

\mathcal{V}_{z,i\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&0% \leq\operatorname{ph}z\leq\pi,\\ \chi(\ell)|z|^{-\ell},&\parbox[t]{224.03743pt}{$-\tfrac{1}{2}\pi\leq% \operatorname{ph}z\leq 0$ or $\pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi$,}\\ 2\chi(\ell)|\Im z|^{-\ell},&\parbox[t]{224.03743pt}{$-\pi<\operatorname{ph}z% \leq-\tfrac{1}{2}\pi$ or $\tfrac{3}{2}\pi\leq\operatorname{ph}z<2\pi$,}\end{cases}

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation, $z$ : complex variable and $\chi(\ell)$
Referenced by:: §10.17(iv)
Permalink:: http://dlmf.nist.gov/10.17.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §10.17(iv), §10.17 and Ch.10

… ►

10.17.18

R_{m,\ell}^{\pm}(\nu,z)=O\left(e^{-2|z|}z^{-m}\right),

|\operatorname{ph}\left(ze^{\mp\frac{1}{2}\pi i}\right)|\leq\pi

ⓘ

Defines:: $R_{\ell}^{\pm}(\nu,z)$ : remainder (locally)
Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $m$ : integer, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.17.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §10.17(v), §10.17 and Ch.10

…

32: 13.29 Methods of Computation

… ►For

M\left(a,b,z\right)

and

M_{\kappa,\mu}\left(z\right)

this means that in the sector

|\operatorname{ph}z|\leq\pi

we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). ►For

U\left(a,b,z\right)

and

W_{\kappa,\mu}\left(z\right)

we may integrate along outward rays from the origin in the sectors

\tfrac{1}{2}\pi<|\operatorname{ph}z|<\tfrac{3}{2}\pi

, with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). In the sector

|\operatorname{ph}z|<\tfrac{1}{2}\pi

the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). On the rays

\operatorname{ph}z=\pm\tfrac{1}{2}\pi

, integration can proceed in either direction. … ►

13.29.6

w(n)={\left(a\right)_{n}}U\left(n+a,b,z\right),

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.29.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

…

33: 15.16 Products

… ►

15.16.1

F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z% \right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}% \right)_{s}}}A_{s}z^{s},

|z|<1

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/15.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

… ►

15.16.3

F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{% \left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z% \zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),

|z|<1

|\zeta|<1

|z+\zeta-z\zeta|<1

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

… ►

15.16.5

F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F% \left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+F\left({% \frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({-\frac% {1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-F\left({\frac{1}{2}+% \lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-% \lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\Gamma\left(1+\lambda% +\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma\left(\lambda+\mu+\nu+\frac{3}{% 2}\right)\Gamma\left(\frac{1}{2}+\nu\right)},

|\operatorname{ph}z|<\pi

|\operatorname{ph}\left(1-z\right)|<\pi

ⓘ

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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/15.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16, §15.16 and Ch.15

…

34: 16.2 Definition and Analytic Properties

… ►

16.2.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum_{k% =0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{{% \left(b_{1}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}}\frac{z^{k}}{k!}.

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §10.16, §10.16, §10.39, §16.2(iv), §16.2(ii), §16.2(iii), §16.2(iii), §16.2(iii), §16.5, §18.17(v), §18.17(vi), §18.17(vii), §18.20(ii), §18.26(i), §18.30(i), §18.38(ii)
Permalink:: http://dlmf.nist.gov/16.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(i), §16.2 and Ch.16

… ►The branch obtained by introducing a cut from

1

+\infty

on the real axis, that is, the branch in the sector

|\operatorname{ph}\left(1-z\right)|\leq\pi

, is the principal branch (or principal value) of

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

; compare §4.2(i). … ►

16.2.3

{{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(% \mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}% \left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/16.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►

16.2.4

\sum_{k=0}^{m}\frac{{\left(\mathbf{a}\right)_{k}}}{{\left(\mathbf{b}\right)_{k% }}}\frac{z^{k}}{k!}=\frac{{\left(\mathbf{a}\right)_{m}}z^{m}}{{\left(\mathbf{b% }\right)_{m}}m!}{{}_{q+2}F_{p}}\left({-m,1,1-m-\mathbf{b}\atop 1-m-\mathbf{a}}% ;\frac{(-1)^{p+q+1}}{z}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/16.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

… ►

16.2.5

{{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{{{}_{p}F_{% q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\Gamma% \left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right)}=\sum_{k=0}^{\infty}% \frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{\Gamma\left(b_{1% }+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{k!};

ⓘ

Defines:: ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ 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
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.2(v)
Permalink:: http://dlmf.nist.gov/16.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.2(v), §16.2 and Ch.16

…

35: 6.18 Methods of Computation

… ►Also, other ranges of

\operatorname{ph}z

can be covered by use of the continuation formulas of §6.4. … ►Zeros of

\operatorname{Ci}\left(x\right)

and

\operatorname{si}\left(x\right)

can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

36: 6.2 Definitions and Interrelations

… ►

6.2.2

E_{1}\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{e^{-t}}{t+z}\,\mathrm{d}t,

|\operatorname{ph}z|<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable
A&S Ref:: 5.1.1 (in modified form)
Referenced by:: §2.5(iii), §6.18(i), §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(i), §6.2 and Ch.6

… ►

6.2.10

\operatorname{si}\left(z\right)=-\int_{z}^{\infty}\frac{\sin t}{t}\,\mathrm{d}% t=\operatorname{Si}\left(z\right)-\tfrac{1}{2}\pi.

ⓘ

Defines:: $\operatorname{si}\left(\NVar{z}\right)$ : sine integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable
A&S Ref:: 5.2.5 (in modified form) 5.2.26 (in modified form)
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(ii), §6.2 and Ch.6

… ►

6.2.17

\mathrm{f}\left(z\right)=\phantom{+}\operatorname{Ci}\left(z\right)\sin z-% \operatorname{si}\left(z\right)\cos z,

ⓘ

Defines:: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals
Symbols:: $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.6
Referenced by:: §6.4, §6.5
Permalink:: http://dlmf.nist.gov/6.2.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(iii), §6.2 and Ch.6

►

6.2.18

\mathrm{g}\left(z\right)=-\operatorname{Ci}\left(z\right)\cos z-\operatorname{% si}\left(z\right)\sin z.

ⓘ

Defines:: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals
Symbols:: $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.7
Referenced by:: §6.4, §6.5
Permalink:: http://dlmf.nist.gov/6.2.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(iii), §6.2 and Ch.6

…

37: 2.6 Distributional Methods

… ►To derive an asymptotic expansion of

\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)

for large values of

|z|

, with

|\operatorname{ph}z|<\pi

, we assume that

f(t)

possesses an asymptotic expansion of the form … ►

2.6.13

\left\langle t^{-s-\alpha},\phi\right\rangle=\frac{1}{{\left(\alpha\right)_{s}% }}\int_{0}^{\infty}t^{-\alpha}\phi^{(s)}(t)\,\mathrm{d}t,

\phi\in\mathcal{T}

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §2.6(ii), §2.6 and Ch.2

… ►The expansion (2.6.7) follows immediately from (2.6.27) with

z=x

and

f(t)=(1+t)^{-(1/3)}

; its region of validity is

|\operatorname{ph}x|\leq\pi-\delta

(

<\pi

). …

38: 17.6 ${{}_{2}\phi_{1}}$ Function

… ►

17.6.2

{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,\ifrac{cq^{n}}{a}\right)=\frac{\left% (c/a;q\right)_{n}}{\left(c;q\right)_{n}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Referenced by:: §17.6(i)
Permalink:: http://dlmf.nist.gov/17.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.3

{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,q\right)=\frac{a^{n}\left(c/a;q% \right)_{n}}{\left(c;q\right)_{n}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.5

{{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{% \infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q% \right)_{\infty}},

|b|>|q|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

17.6.8

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(\ifrac{abz}{c};q% \right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,c/b% \atop c};q,\ifrac{abz}{c}\right),

|z|<1,|abz|<|c|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

… ►where

|z|<1

|\operatorname{ph}\left(-z\right)|<\pi

, and the contour of integration separates the poles of

\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}/\sin\left(\pi\zeta\right)

from those of

1/\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}

, and the infimum of the distances of the poles from the contour is positive. …

39: Bibliography T

… ►

Go. Torres-Vega, J. D. Morales-Guzmán, and A. Zúñiga-Segundo (1998) Special functions in phase space: Mathieu functions. J. Phys. A 31 (31), pp. 6725–6739.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (Bing Hong Wang), ZentralBlatt
Referenced by:: 8th item.
Encodings:: BibTeX

… ►

S. Tsujimoto, L. Vinet, and A. Zhedanov (2012) Dunkl shift operators and Bannai-Ito polynomials. Adv. Math. 229 (4), pp. 2123–2158.

ⓘ

External Links:: ISSN 0001-8708, Link, MathReview (Juan José Moreno-Balcázar), ZentralBlatt
Referenced by:: §18.28(xi).
Encodings:: BibTeX

…

40: 13.14 Definitions and Basic Properties

… ►

13.14.20

M_{\kappa,\mu}\left(z\right)\sim\ifrac{\Gamma\left(1+2\mu\right)e^{\frac{1}{2}% z}z^{-\kappa}}{\Gamma\left(\tfrac{1}{2}+\mu-\kappa\right)},

\left|\operatorname{ph}z\right|\leq\frac{1}{2}\pi-\delta

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $z$ : complex variable and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/13.14.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §13.14(iv), §13.14 and Ch.13

… ►

13.14.21

W_{\kappa,\mu}\left(z\right)\sim e^{-\frac{1}{2}z}z^{\kappa},

\left|\operatorname{ph}z\right|\leq\frac{3}{2}\pi-\delta

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $z$ : complex variable and $\delta$ : small positive constant
Referenced by:: §33.21(i)
Permalink:: http://dlmf.nist.gov/13.14.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §13.14(iv), §13.14 and Ch.13

… ►

W_{-\kappa,\mu}\left(e^{-\pi\mathrm{i}}z\right)

-\tfrac{1}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{3}{2}\pi

… ►

W_{-\kappa,\mu}\left(e^{\pi\mathrm{i}}z\right)

-\tfrac{3}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{1}{2}\pi

►A fundamental pair of solutions that is numerically satisfactory in the sector

|\operatorname{ph}{z}|\leq\pi

near the origin is …