Jacobi fraction (J-fraction)

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1—10 of 217 matching pages

►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). … ►

Jacobi on Other Intervals

…

3: 3.10 Continued Fractions

§3.10 Continued Fractions

… ►

Stieltjes Fractions

… ►

Jacobi Fractions

… ►is called a Jacobi fraction ( $J$ -fraction). …For the same function

f(z)

, the convergent

C_{n}

of the Jacobi fraction (3.10.11) equals the convergent

C_{2n}

of the Stieltjes fraction (3.10.6). …

4: 20.11 Generalizations and Analogs

… ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►This is Jacobi’s inversion problem of §20.9(ii). … ►Each provides an extension of Jacobi’s inversion problem. … ►For

m=1,2,3,4

n=1,2,3,4

, and

m\neq n

, define twelve combined theta functions

\varphi_{m,n}\left(z,q\right)

by …

5: 22.8 Addition Theorems

… ►

22.8.1

\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},

ⓘ

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, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.1
Referenced by:: §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

… ►

22.8.14

\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn% }u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},

ⓘ

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, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(ii), §22.8 and Ch.22

►

22.8.15

\operatorname{cn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v-\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{sn% }u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},

ⓘ

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, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(ii), §22.8 and Ch.22

… ►

22.8.17

\operatorname{dn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}u-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}v}{\operatorname{sn% }u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},

ⓘ

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, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(ii), §22.8 and Ch.22

… ►

22.8.23

\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.

ⓘ

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, $\det$ : determinant, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

…

6: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.2

2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{% \sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(% \sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : 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, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.12, §22.12
Permalink:: http://dlmf.nist.gov/22.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

… ►

22.12.8

2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=% -\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : 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, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

… ►

22.12.11

2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m}}{t-m-n\tau}\right),

ⓘ

Symbols:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : 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, $\sin\NVar{z}$ : sine function, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.12.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

… ►

22.12.13

2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}% \frac{\pi}{\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)% ^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : 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, $\tan\NVar{z}$ : tangent function, $k$ : modulus and $\tau$ : ratio
Referenced by:: §22.12
Permalink:: http://dlmf.nist.gov/22.12.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.12 and Ch.22

7: 12.6 Continued Fraction

§12.6 Continued Fraction

►For a continued-fraction expansion of the ratio

\ifrac{U\left(a,x\right)}{U\left(a-1,x\right)}

see Cuyt et al. (2008, pp. 340–341).

8: 22.6 Elementary Identities

… ►

22.6.2

1+{\operatorname{cs}}^{2}\left(z,k\right)=k^{2}+{\operatorname{ds}}^{2}\left(z% ,k\right)={\operatorname{ns}}^{2}\left(z,k\right),

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.9.4
Permalink:: http://dlmf.nist.gov/22.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(i), §22.6 and Ch.22

… ►

22.6.5

\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{% \operatorname{sn}}^{4}\left(z,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, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.18.1
Permalink:: http://dlmf.nist.gov/22.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

22.6.8

\operatorname{cd}\left(2z,k\right)=\frac{{\operatorname{cd}}^{2}\left(z,k% \right)-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right){\operatorname{% nd}}^{2}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(% z,k\right)},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

►

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

► ►►

$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
…

► …

9: 22.13 Derivatives and Differential Equations

… ►

22.13.1

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{sn}}^{2}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{sn}}^{2}\left(z,k\right)\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Referenced by:: §22.13(iii), §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

►

22.13.2

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{% 2}={\left(1-{\operatorname{cn}}^{2}\left(z,k\right)\right)}{\left({k^{\prime}}% ^{2}+k^{2}{\operatorname{cn}}^{2}\left(z,k\right)\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

►

22.13.3

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

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.7

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{% 2}=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({\operatorname{% dc}}^{2}\left(z,k\right)-k^{2}\right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.10

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ns}\left(z,k\right)\right)^{% 2}=\left({\operatorname{ns}}^{2}\left(z,k\right)-k^{2}\right)\left({% \operatorname{ns}}^{2}\left(z,k\right)-1\right),

ⓘ

Symbols:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

…

10: 22.21 Tables

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates

\operatorname{sn}\left(Kx,k\right)

\operatorname{cn}\left(Kx,k\right)

\operatorname{dn}\left(Kx,k\right)

\operatorname{am}\left(Kx,k\right)

\mathcal{E}\left(Kx,k\right)

for

\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}

and

x=0\left(\tfrac{1}{90}\right)1

to 12D, or 12 decimals of a radian in the case of

\operatorname{am}\left(Kx,k\right)

. … ►Lawden (1989, pp. 280–284 and 293–297) tabulates

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

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

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

\mathcal{E}\left(x,k\right)

\mathrm{Z}\left(x|k\right)

to 5D for

k=0.1(.1)0.9

x=0(.1)X

, where

X

ranges from 1. … ►Zhang and Jin (1996, p. 678) tabulates

\operatorname{sn}\left(Kx,k\right)

\operatorname{cn}\left(Kx,k\right)

\operatorname{dn}\left(Kx,k\right)

for

k=\frac{1}{4},\frac{1}{2}

and

x=0(.1)4

to 7D. …

Jacobi fraction (J-fraction)

1—10 of 217 matching pages

1: 22.16 Related Functions

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

§22.16(ii) Jacobi’s Epsilon Function

Integral Representations

§22.16(iii) Jacobi’s Zeta Function

Properties

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals

3: 3.10 Continued Fractions

§3.10 Continued Fractions

Stieltjes Fractions

Jacobi Fractions

4: 20.11 Generalizations and Analogs

5: 22.8 Addition Theorems

6: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

7: 12.6 Continued Fraction

§12.6 Continued Fraction

8: 22.6 Elementary Identities

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

9: 22.13 Derivatives and Differential Equations

10: 22.21 Tables

§22.21 Tables

Jacobi fraction (J-fraction)

1—10 of 217 matching pages

1: 22.16 Related Functions

§22.16(i) Jacobi’s Amplitude ( am ) Function

§22.16(ii) Jacobi’s Epsilon Function

Integral Representations

§22.16(iii) Jacobi’s Zeta Function

Properties

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals

3: 3.10 Continued Fractions

§3.10 Continued Fractions

Stieltjes Fractions

Jacobi Fractions

4: 20.11 Generalizations and Analogs

5: 22.8 Addition Theorems

6: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

7: 12.6 Continued Fraction

§12.6 Continued Fraction

8: 22.6 Elementary Identities

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

9: 22.13 Derivatives and Differential Equations

10: 22.21 Tables

§22.21 Tables

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function