DLMF: Untitled Document

§22.2 Definitions

►The nome

q

is given in terms of the modulus

k

by … ►

22.2.3

\zeta=\frac{\pi z}{2K\left(k\right)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►

22.2.11

\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p}\left(z\middle|\tau\right)}% {\theta_{q}\left(z\middle|\tau\right)},

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $q$ : nome, $z$ : complex, $k$ : modulus and $\tau$ : ratio
A&S Ref:: 16.36.3
Referenced by:: §22.2
Permalink:: http://dlmf.nist.gov/22.2.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2, §22.2 and Ch.22

… ►

22.2.12

\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{K\left(k\right)},

ⓘ

Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.2
Permalink:: http://dlmf.nist.gov/22.2.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2, §22.2 and Ch.22

…

§22.17 Moduli Outside the Interval [0,1]

►

§22.17(i) Real or Purely Imaginary Moduli

… ►

22.17.2

\operatorname{sn}\left(z,1/k\right)=k\operatorname{sn}\left(z/k,k\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.2
Permalink:: http://dlmf.nist.gov/22.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

§22.17(ii) Complex Moduli

… ►For proofs of these results and further information see Walker (2003).

§9.8 Modulus and Phase

►

§9.8(i) Definitions

… ►

§9.8(ii) Identities

… ►

§9.8(iii) Monotonicity

… ►

Table 4.30.1: Hyperbolic functions: interrelations. …

► ►►►►►►►

	$\sinh\theta=a$	$\cosh\theta=a$	$\tanh\theta=a$	$\operatorname{csch}\theta=a$	$\operatorname{sech}\theta=a$	$\coth\theta=a$
$\sinh\theta$	$a$	$(a^{2}-1)^{1/2}$	$a(1-a^{2})^{-1/2}$	$a^{-1}$	$a^{-1}(1-a^{2})^{1/2}$	$(a^{2}-1)^{-1/2}$
$\cosh\theta$	$(1+a^{2})^{1/2}$	$a$	$(1-a^{2})^{-1/2}$	$a^{-1}(1+a^{2})^{1/2}$	$a^{-1}$	$a(a^{2}-1)^{-1/2}$
$\tanh\theta$	$a(1+a^{2})^{-1/2}$	$a^{-1}(a^{2}-1)^{1/2}$	$a$	$(1+a^{2})^{-1/2}$	$(1-a^{2})^{1/2}$	$a^{-1}$
$\operatorname{csch}\theta$	$a^{-1}$	$(a^{2}-1)^{-1/2}$	$a^{-1}(1-a^{2})^{1/2}$	$a$	$a(1-a^{2})^{-1/2}$	$(a^{2}-1)^{1/2}$
$\operatorname{sech}\theta$	$(1+a^{2})^{-1/2}$	$a^{-1}$	$(1-a^{2})^{1/2}$	$a(1+a^{2})^{-1/2}$	$a$	$a^{-1}(a^{2}-1)^{1/2}$
…

►

… ►For the modulus functions

M\left(x\right)

and

N\left(x\right)

see §10.68(i) with

\nu=0

. … ►

See accompanying text — Figure 10.62.3: $e^{-x/\sqrt{2}}\operatorname{ber}x$ , $e^{-x/\sqrt{2}}\operatorname{bei}x$ , $e^{-x/\sqrt{2}}M\left(x\right)$ , $0\leq x\leq 8$ . Magnify

►

… ►

17.10.1

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.3

{{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.4

{{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.5

\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -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/17.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.6

\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \*{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{3}}{bcdefghk}\right)=\frac{\left(q,q% /(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q\right)_{\infty}}{% \left(gh,gk,h/g,k/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{\infty}}\*{{}_{10}\phi% _{9}}\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg% /f,qg/h,qg/k};q,\frac{q^{3}}{bcdefghk}\right)+\operatorname{idem}\left(g;h,k% \right).

ⓘ

Symbols:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $k$ : nonnegative integer and $q$ : complex base
Referenced by:: Erratum (V1.2.4) for Equation (17.10.6)
Permalink:: http://dlmf.nist.gov/17.10.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.2.4):: In the numerator of the argument of the basic bilateral hypergeometric function ${{}_{10}\psi_{10}}$ and in the numerator of the arguments of the basic hypergeometric ${{}_{10}\phi_{9}}$ functions, we replaced $q^{2}$ by $q^{3}$ . We also added a missing factor $1/\left(k/g;q\right)_{\infty}$ in the first term on the right-hand side.
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.9.2

{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{n% }}{\left(c;q\right)_{n}}b^{n}{{}_{3}\phi_{1}}\left({q^{-n},b,q/z\atop bq^{1-n}% /c};q,z/c\right),

ⓘ

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, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: Erratum (V1.0.14) for Equation (17.9.2)
Permalink:: http://dlmf.nist.gov/17.9.E2
Encodings:: pMML, png, TeX
Correction (effective with 1.0.14):: The entry $q/c$ in the first row of ${{}_{3}\phi_{1}}\left({q^{-n},b,q/c\atop bq^{1-n}/c};q,z/c\right)$ was replaced by $q/z$ .
Suggested 2016-08-30 by Xinrong Ma
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

… ►

17.9.12

{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c,cq/a,q/d;q\right)_{\infty}}{\left(e,cq/d,q/a,e/(bc);q\right)_{\infty}}{{}% _{3}\phi_{2}}\left({c,d/a,cq/e\atop cq/a,bcq/e};q,\frac{bq}{d}\right)-\frac{% \left(q/d,eq/d,b,c,d/a,de/(bcq),bcq^{2}/(de);q\right)_{\infty}}{\left(d/q,e,bq% /d,cq/d,q/a,e/(bc),bcq/e;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({aq/d,bq/d,cq% /d\atop q^{2}/d,eq/d};q,\frac{de}{abc}\right),

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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/17.9.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

►

17.9.13

{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{% \infty}}{\left(d,e,bc/e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({e/b,% e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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
Referenced by:: (17.9.3_5)
Permalink:: http://dlmf.nist.gov/17.9.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

… ►

17.9.14

{{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

… ►

17.9.16

{{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^{2}q^{2}}{% bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q\right)_{\infty}}{\left(% aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}{{}_{4}\phi_{3}}\left({aq/(bc),d,e,f% \atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(% bdef),a^{2}q^{2}/(cdef);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2% }q^{2}/(bcdef),def/(aq);q\right)_{\infty}}\*{{}_{4}\phi_{3}}\left({aq/(de),aq/% (df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2% }/(def)};q,q\right).

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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/17.9.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

…

§10.68 Modulus and Phase Functions

►

§10.68(i) Definitions

… ►

§10.68(ii) Basic Properties

… ►

§10.68(iii) Asymptotic Expansions for Large Argument

… ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …

… ►

17.8.4

{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},

\left|qa\right|<\left|bc\right|

,

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.2.1) for Equation (17.8.4)
Permalink:: http://dlmf.nist.gov/17.8.E4
Encodings:: pMML, png, TeX
Correction (effective with 1.2.1):: The constraint $\left|qa\right|<\left|bc\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.5

{{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{% \left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,q/c,q/d,q/(bcd);q% \right)_{\infty}},

\left|q\right|<\left|bcd\right|

,

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.2.1) for Equation (17.8.5)
Permalink:: http://dlmf.nist.gov/17.8.E5
Encodings:: pMML, png, TeX
Correction (effective with 1.2.1):: The constraint $\left|q\right|<\left|bcd\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.6

{{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},

\left|qa^{\frac{3}{2}}\right|<\left|bcd\right|

,

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.2.1) for Equation (17.8.6)
Permalink:: http://dlmf.nist.gov/17.8.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.2.1):: The constraint $\left|qa^{\frac{3}{2}}\right|<\left|bcd\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.7

{{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}},

\left|qa^{2}\right|<\left|bcde\right|

.

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.2.1) for Equation (17.8.7)
Permalink:: http://dlmf.nist.gov/17.8.E7
Encodings:: pMML, png, TeX
Correction (effective with 1.2.1):: The constraint $\left|qa^{2}\right|<\left|bcde\right|$ was added.
See also:: Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.8

{{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),

\left|cq^{2}\right|<\left|b^{2}\right|

.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Source:: Verma and Jain (1983, (3.9))
Referenced by:: §17.6(i), §17.8, Erratum (V1.1.0) for Additions, Erratum (V1.2.1) for Equation (17.8.8)
Permalink:: http://dlmf.nist.gov/17.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.8, §17.8 and Ch.17

…

… ►For example, at

z=K+iK^{\prime}

,

\operatorname{sn}\left(z,k\right)=1/k

,

\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0

. (The modulus

k

is suppressed throughout the table.) … ►For example,

\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}

. … ►

§22.5(ii) Limiting Values of $k$

… ►For values of

K,K^{\prime}

when

k^{2}=\frac{1}{2}

(lemniscatic case) see §23.5(iii), and for

k^{2}=e^{\mathrm{i}\pi/3}

(equianharmonic case) see §23.5(v). …

modulus

1—10 of 537 matching pages

1: 22.2 Definitions

§22.2 Definitions

2: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

§22.17(i) Real or Purely Imaginary Moduli

§22.17(ii) Complex Moduli

3: 9.8 Modulus and Phase

§9.8 Modulus and Phase

§9.8(i) Definitions

§9.8(ii) Identities

§9.8(iii) Monotonicity

4: 4.30 Elementary Properties

5: 10.62 Graphs

6: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

7: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

8: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

§10.68(i) Definitions

§10.68(ii) Basic Properties

§10.68(iii) Asymptotic Expansions for Large Argument

9: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

10: 22.5 Special Values

§22.5(ii) Limiting Values of $k$

modulus

1—10 of 537 matching pages

1: 22.2 Definitions

§22.2 Definitions

2: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

§22.17(i) Real or Purely Imaginary Moduli

§22.17(ii) Complex Moduli

3: 9.8 Modulus and Phase

§9.8 Modulus and Phase

§9.8(i) Definitions

§9.8(ii) Identities

§9.8(iii) Monotonicity

4: 4.30 Elementary Properties

5: 10.62 Graphs

6: 17.10 Transformations of ψ r r Functions

7: 17.9 Further Transformations of ϕ r r + 1 Functions

8: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

§10.68(i) Definitions

§10.68(ii) Basic Properties

§10.68(iii) Asymptotic Expansions for Large Argument

9: 17.8 Special Cases of ψ r r Functions

10: 22.5 Special Values

§22.5(ii) Limiting Values of k

6: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

7: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

9: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

§22.5(ii) Limiting Values of $k$