Jacobi fraction (J-fraction)

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21: 22.7 Landen Transformations

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22.7.2

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.3

\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.3
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.4

\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.4
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

… ►

22.7.6

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.2
Permalink:: http://dlmf.nist.gov/22.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

… ►

22.7.8

\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.4
Permalink:: http://dlmf.nist.gov/22.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

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22: 18.14 Inequalities

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Jacobi

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Jacobi

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Jacobi

… ►

Szegő–Szász Inequality

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Jacobi

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23: 27.9 Quadratic Characters

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27.9.3

(p|q)(q|p)=(-1)^{(p-1)(q-1)/4}.

ⓘ

Symbols:: $(\NVar{n}|\NVar{p})$ : Legendre symbol, $p$ : odd prime and $q$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.9 and Ch.27

►If an odd integer

P

has prime factorization

P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}

, then the Jacobi symbol

(n|P)

is defined by

(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}

, with

(n|1)=1

. The Jacobi symbol

(n|P)

is a Dirichlet character (mod

P

). …

24: 22.2 Definitions

… ►

22.2.4

\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►

22.2.5

\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nc}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►

22.2.6

\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}% \left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nd}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §22.20(ii)
Permalink:: http://dlmf.nist.gov/22.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►

22.2.7

\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}}^{2}\left(0,q\right)}{% \theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(% \zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}% \left(z,k\right)},

ⓘ

Defines:: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►

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

. …

25: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. … ►

Table 18.6.1: Classical OP’s: symmetry and special values.

► ►►

$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
…

► ►

§18.6(ii) Limits to Monomials

►

18.6.2

\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}\right)^{n},

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.6(ii), §18.6 and Ch.18

►

18.6.3

\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}\right)^{n},

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.6(ii), §18.6 and Ch.18

…

26: 20.15 Tables

… ►This reference gives

\theta_{j}\left(x,q\right)

j=1,2,3,4

, and their logarithmic

x

-derivatives to 4D for

x/\pi=0(.1)1

\alpha=0(9^{\circ})90^{\circ}

, where

\alpha

is the modular angle given by ►

20.15.1

\sin\alpha={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)=k.

ⓘ

Defines:: $\alpha$ : modular angle (locally)
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sin\NVar{z}$ : sine function and $q$ : nome
Referenced by:: §20.15, §20.15
Permalink:: http://dlmf.nist.gov/20.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.15 and Ch.20

►Spenceley and Spenceley (1947) tabulates

\theta_{1}\left(x,q\right)/\theta_{2}\left(0,q\right)

\theta_{2}\left(x,q\right)/\theta_{2}\left(0,q\right)

\theta_{3}\left(x,q\right)/\theta_{4}\left(0,q\right)

\theta_{4}\left(x,q\right)/\theta_{4}\left(0,q\right)

to 12D for

u=0(1^{\circ})90^{\circ}

\alpha=0(1^{\circ})89^{\circ}

, where

u=2x/(\pi{\theta_{3}}^{2}\left(0,q\right))

and

\alpha

is defined by (20.15.1), together with the corresponding values of

\theta_{2}\left(0,q\right)

and

\theta_{4}\left(0,q\right)

. ►Lawden (1989, pp. 270–279) tabulates

\theta_{j}\left(x,q\right)

j=1,2,3,4

, to 5D for

x=0(1^{\circ})90^{\circ}

q=0.1(.1)0.9

, and also

q

to 5D for

k^{2}=0(.01)1

. ►Tables of Neville’s theta functions

\theta_{s}\left(x,q\right)

\theta_{c}\left(x,q\right)

\theta_{d}\left(x,q\right)

\theta_{n}\left(x,q\right)

(see §20.1) and their logarithmic

x

-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for

\varepsilon,\alpha=0(5^{\circ})90^{\circ}

, where (in radian measure)

\varepsilon=x/{\theta_{3}}^{2}\left(0,q\right)=\pi x/(2K\left(k\right))

, and

\alpha

is defined by (20.15.1). …

27: 20.8 Watson’s Expansions

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20.8.1

\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : integer, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.8 and Ch.20

…

28: 22.17 Moduli Outside the Interval [0,1]

… ►

22.17.1

\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),

ⓘ

Symbols:: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

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

►

22.17.3

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

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.3
Permalink:: http://dlmf.nist.gov/22.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.4

\operatorname{dn}\left(z,1/k\right)=\operatorname{cn}\left(z/k,k\right).

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.4
Permalink:: http://dlmf.nist.gov/22.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

22.17.7

\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $z$ : complex, $k$ : modulus, $k_{1}$ : change of variable and $k_{1}^{\prime}$ : change of variable
A&S Ref:: 16.10.3 (modified)
Permalink:: http://dlmf.nist.gov/22.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

…

29: 18.4 Graphics

… ►

See accompanying text — Figure 18.4.1: Jacobi polynomials $P^{(1.5,-0.5)}_{n}\left(x\right)$ , $n=1,2,3,4,5$ . Magnify

►

…

30: 22.15 Inverse Functions

… ►are denoted respectively by ►

\xi=\operatorname{arcsn}\left(x,k\right),

►

\eta=\operatorname{arccn}\left(x,k\right),

►

\zeta=\operatorname{arcdn}\left(x,k\right).

… ►Equations (22.15.1) and (22.15.4), for

\operatorname{arcsn}\left(x,k\right)

, are equivalent to (22.15.12) and also to …