Jacobi%20nome

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►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: 20.7 Identities

… ►

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

… ►

§20.7(iv) Reduction Formulas for Products

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

►

20.7.34

\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $q$ : nome
Referenced by:: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/20.7.E34
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. See Walker (2012).
See also:: Annotations for §20.7(ix), §20.7 and Ch.20

…

4: 22.2 Definitions

§22.2 Definitions

►The nome

q

is given in terms of the modulus

k

by … ►

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

… ►

22.2.7

\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

. …

5: 20.11 Generalizations and Analogs

… ►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 … ►

20.11.9

\varphi_{m,n}\left(z,q\right)=\varphi_{m,1}\left(z,q\right)\varphi_{1,n}\left(% z,q\right)=\frac{1}{\varphi_{n,m}\left(z,q\right)}=\frac{\varphi_{m,1}\left(z,% q\right)}{\varphi_{n,1}\left(z,q\right)}=\frac{\varphi_{1,n}\left(z,q\right)}{% \varphi_{1,m}\left(z,q\right)}.

ⓘ

Symbols:: $\varphi_{\NVar{n},\NVar{m}}\left(\NVar{z},\NVar{q}\right)$ : combined theta function, $m$ : integer, $n$ : integer, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §20.11(v), §20.11 and Ch.20

…

6: 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

… ►

20.4.9

\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}=-1-8\sum_{n=1}% ^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}},

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(ii), §20.4 and Ch.20

…

7: 22.21 Tables

§22.21 Tables

… ►Curtis (1964b) tabulates

\operatorname{sn}\left(mK/n,k\right)

\operatorname{cn}\left(mK/n,k\right)

\operatorname{dn}\left(mK/n,k\right)

for

n=2(1)15

m=1(1)n-1

, and

q

(not

k

)

=0(.005)0.35

to 20D. ►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. …

8: 22.1 Special Notation

… ► ►►►

$x, y$	real variables.
…
$q$	nome. $0\leq q<1$ except in §22.17; see also §20.1.
…

… ►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). …

9: 22.11 Fourier and Hyperbolic Series

… ►Throughout this section

q

and

\zeta

are defined as in §22.2. ►If

q\exp\left(2|\Im\zeta|\right)<1

, then … ►Next, if

q\exp\left(|\Im\zeta|\right)<1

, then … ►Similar expansions for

{\operatorname{cn}}^{2}\left(z,k\right)

and

{\operatorname{dn}}^{2}\left(z,k\right)

follow immediately from (22.6.1). … ►Again, similar expansions for

{\operatorname{cn}}^{2}\left(z,k\right)

and

{\operatorname{dn}}^{2}\left(z,k\right)

may be derived via (22.6.1). …

10: 20.8 Watson’s Expansions

… ►

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

…

Jacobi%20nome

1—10 of 215 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

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals

3: 20.7 Identities

§20.7(iv) Reduction Formulas for Products

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

4: 22.2 Definitions

§22.2 Definitions

5: 20.11 Generalizations and Analogs

6: 20.4 Values at $z$ = 0

Jacobi’s Identity

7: 22.21 Tables

§22.21 Tables

8: 22.1 Special Notation

9: 22.11 Fourier and Hyperbolic Series

10: 20.8 Watson’s Expansions

Jacobi%20nome

1—10 of 215 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

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals

3: 20.7 Identities

§20.7(iv) Reduction Formulas for Products

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

4: 22.2 Definitions

§22.2 Definitions

5: 20.11 Generalizations and Analogs

6: 20.4 Values at z = 0

Jacobi’s Identity

7: 22.21 Tables

§22.21 Tables

8: 22.1 Special Notation

9: 22.11 Fourier and Hyperbolic Series

10: 20.8 Watson’s Expansions

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

6: 20.4 Values at $z$ = 0