Jacobi fraction (J-fraction)

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11—20 of 217 matching pages

12: 22.4 Periods, Poles, and Zeros

… ►For example, the poles of

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

, abbreviated as

\operatorname{sn}

in the following tables, are at

z=2mK+(2n+1)iK^{\prime}

. … ►Then: (a) In any lattice unit cell

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

has a simple zero at

z=\mbox{p}

and a simple pole at

z=\mbox{q}

. (b) The difference between p and the nearest q is a half-period of

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

. This half-period will be plus or minus a member of the triple

{K,iK^{\prime},K+iK^{\prime}}

; the other two members of this triple are quarter periods of

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

. … ►For example,

\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)

. …

13: 20.4 Values at $z$ = 0

… ►

20.4.1

\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)=\theta_{4}'\left(0,q\right)=0,

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4 and Ch.20

… ►

Jacobi’s Identity

►

20.4.6

\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
A&S Ref:: 16.28.6
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4(i), §20.4 and Ch.20

… ►

20.4.7

\theta_{1}''\left(0,q\right)=\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(% 0,q\right)=\theta_{4}'''\left(0,q\right)=0.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(ii), §20.4 and Ch.20

… ►

20.4.12

\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=\frac{\theta% _{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}+\frac{\theta_{3}''\left(0,% q\right)}{\theta_{3}\left(0,q\right)}+\frac{\theta_{4}''\left(0,q\right)}{% \theta_{4}\left(0,q\right)}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(ii), §20.4 and Ch.20

14: 20.7 Identities

… ►

20.7.6

{\theta_{4}}^{2}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

►

20.7.7

{\theta_{4}}^{2}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

►

20.7.8

{\theta_{4}}^{2}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

►

20.7.9

{\theta_{4}}^{2}\left(0,q\right)\theta_{4}\left(w+z,q\right)\theta_{4}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

… ►

20.7.10

\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(iii), §20.7 and Ch.20

…

15: 18.30 Associated OP’s

… ►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. … ►

§18.30(i) Associated Jacobi Polynomials

… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). … ►See Ismail (2009, p. 46 ), where the

k

th corecursive polynomial is also related to an appropriate continued fraction, given here as its

n

th convergent, …

16: 10.55 Continued Fractions

§10.55 Continued Fractions

►For continued fractions for

\mathsf{j}_{n+1}\left(z\right)/\mathsf{j}_{n}\left(z\right)

and

{\mathsf{i}^{(1)}_{n+1}}\left(z\right)/{\mathsf{i}^{(1)}_{n}}\left(z\right)

see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).

17: 15.9 Relations to Other Functions

… ►

Jacobi

… ►

§15.9(ii) Jacobi Function

… ►The Jacobi transform is defined as …with inverse … …

18: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions

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

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

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

; the nine subsidiary Jacobian elliptic functions

\operatorname{cd}\left(z,k\right)

\operatorname{sd}\left(z,k\right)

\operatorname{nd}\left(z,k\right)

\operatorname{dc}\left(z,k\right)

\operatorname{nc}\left(z,k\right)

\operatorname{sc}\left(z,k\right)

\operatorname{ns}\left(z,k\right)

\operatorname{ds}\left(z,k\right)

\operatorname{cs}\left(z,k\right)

; the amplitude function

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

; Jacobi’s epsilon and zeta functions

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

and

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

. … ►The notation

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

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

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

is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for

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

are

\mathrm{sn}(z\mathpunct{|}m)

and

\mathrm{sn}(z,m)

with

m=k^{2}

; see Abramowitz and Stegun (1964) and Walker (1996). …

19: 18.7 Interrelations and Limit Relations

… ►

Ultraspherical and Jacobi

… ►

Chebyshev, Ultraspherical, and Jacobi

… ►

Legendre, Ultraspherical, and Jacobi

… ►

Jacobi $\to$ Laguerre

… ►

Jacobi $\to$ Hermite

…

20: 22.14 Integrals

… ►

22.14.3

\int\operatorname{dn}\left(x,k\right)\,\mathrm{d}x=\operatorname{Arcsin}\left(% \operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right).

ⓘ

Symbols:: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.3
Permalink:: http://dlmf.nist.gov/22.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

… ►See §22.16(i) for

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

. … ►

22.14.13

\int\frac{\,\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\ln\left(\frac{% \operatorname{sn}\left(x,k\right)}{\operatorname{cn}\left(x,k\right)+% \operatorname{dn}\left(x,k\right)}\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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
Referenced by:: §22.14(iii)
Permalink:: http://dlmf.nist.gov/22.14.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(iii), §22.14 and Ch.22

►

22.14.14

\int\frac{\operatorname{cn}\left(x,k\right)\,\mathrm{d}x}{\operatorname{sn}% \left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-\operatorname{dn}\left(x,k\right% )}{1+\operatorname{dn}\left(x,k\right)}\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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.14.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(iii), §22.14 and Ch.22

►

22.14.15

\int\frac{\operatorname{cn}\left(x,k\right)\,\mathrm{d}x}{{\operatorname{sn}}^% {2}\left(x,k\right)}=-\frac{\operatorname{dn}\left(x,k\right)}{\operatorname{% sn}\left(x,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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus
Referenced by:: §22.14(iii)
Permalink:: http://dlmf.nist.gov/22.14.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(iii), §22.14 and Ch.22

…

Jacobi fraction (J-fraction)

11—20 of 217 matching pages

11: 18.17 Integrals

Jacobi

Jacobi

Jacobi

Jacobi

Jacobi

12: 22.4 Periods, Poles, and Zeros

13: 20.4 Values at z = 0

Jacobi’s Identity

14: 20.7 Identities

15: 18.30 Associated OP’s

§18.30(i) Associated Jacobi Polynomials

16: 10.55 Continued Fractions

§10.55 Continued Fractions

17: 15.9 Relations to Other Functions

Jacobi

§15.9(ii) Jacobi Function

18: 22.1 Special Notation

19: 18.7 Interrelations and Limit Relations

Ultraspherical and Jacobi

Chebyshev, Ultraspherical, and Jacobi

Legendre, Ultraspherical, and Jacobi

Jacobi → Laguerre

Jacobi → Hermite

20: 22.14 Integrals

13: 20.4 Values at $z$ = 0

Jacobi $\to$ Laguerre

Jacobi $\to$ Hermite