Jacobi

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21: 22.17 Moduli Outside the Interval [0,1]

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22.17.1 $$\mathrm{p}\mathrm{q}(z,k)=\mathrm{p}\mathrm{q}(z,-k),$$

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Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: 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

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22.17.2 $$\mathrm{sn}(z,1/k)=k\mathrm{sn}(z/k,k),$$

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Symbols:

$\mathrm{sn}(z,k)$: 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

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22.17.3 $$\mathrm{cn}(z,1/k)=\mathrm{dn}(z/k,k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: 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

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22.17.4 $$\mathrm{dn}(z,1/k)=\mathrm{cn}(z/k,k).$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: 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

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22.17.7 $$\mathrm{cn}(z,\mathrm{i}k)=\mathrm{cd}(z/k_1^{\prime },k_1),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{cn}(z,k)$: 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

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22: 18.4 Graphics

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

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Figure 18.4.2: Jacobi polynomials $P_n^{(1.25,0.75)}\left(x\right)$, $n=7,8$. This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). … Magnify

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23: 22.15 Inverse Functions

… ►are denoted respectively by ►

$\xi =\mathrm{arcsn}(x,k),$

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$\eta =\mathrm{arccn}(x,k),$

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$\zeta =\mathrm{arcdn}(x,k).$

… ►Equations (22.15.1) and (22.15.4), for $\mathrm{arcsn}(x,k)$, are equivalent to (22.15.12) and also to …

24: 22.5 Special Values

… ►For example, at $z=K+\mathrm{i}{K}^{\prime }$, $\mathrm{sn}(z,k)=1/k$, $d\mathrm{sn}(z,k)/dz=0$. … ►Table 22.5.2 gives $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ for other special values of $z$. For example, $\mathrm{sn}(\frac{1}{2}K,k)={(1+k^{\prime })}^{-1/2}$. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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$\mathrm{dn}(z,k)$ $\to$	$1$	$\mathrm{nd}(z,k)$ $\to$	$1$	$\mathrm{sc}(z,k)$ $\to$	$\tan z$	$\mathrm{cs}(z,k)$ $\to$	$\cot z$

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25: 20.1 Special Notation

… ►The main functions treated in this chapter are the theta functions ${\theta }_j\left(z\right|\tau )={\theta }_j(z,q)$ where $j=1,2,3,4$ and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$. When $\tau$ is fixed the notation is often abbreviated in the literature as ${\theta }_j(z)$, or even as simply ${\theta }_j$, it being then understood that the argument is the primary variable. … ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. … ►Jacobi’s original notation: $\mathrm{\Theta }(z|\tau )$, ${\mathrm{\Theta }}_1(z|\tau )$, $\mathrm{H}(z|\tau )$, ${\mathrm{H}}_1(z|\tau )$, respectively, for ${\theta }_4\left(u\right|\tau )$, ${\theta }_3\left(u\right|\tau )$, ${\theta }_1\left(u\right|\tau )$, ${\theta }_2\left(u\right|\tau )$, where $u=z/{\theta }_3^2\left(0\right|\tau )$. … ►Neville’s notation: ${\theta }_s(z|\tau )$, ${\theta }_c(z|\tau )$, ${\theta }_d(z|\tau )$, ${\theta }_n(z|\tau )$, respectively, for ${\theta }_3^2\left(0\right|\tau ){\theta }_1\left(u\right|\tau )/{\theta }_1^{\prime }\left(0\right|\tau )$, ${\theta }_2\left(u\right|\tau )/{\theta }_2\left(0\right|\tau )$, ${\theta }_3\left(u\right|\tau )/{\theta }_3\left(0\right|\tau )$, ${\theta }_4\left(u\right|\tau )/{\theta }_4\left(0\right|\tau )$, where again $u=z/{\theta }_3^2\left(0\right|\tau )$. …

26: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. … ►For fixed $\tau$, each ${\theta }_j\left(z\right|\tau )$ is an entire function of $z$ with period $2\pi$; ${\theta }_1\left(z\right|\tau )$ is odd in $z$ and the others are even. For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. … ►For $m,n\in \mathbb{Z}$, the $z$-zeros of ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, are $(m+n\tau )\pi$, $(m+\frac{1}{2}+n\tau )\pi$, $(m+\frac{1}{2}+(n+\frac{1}{2})\tau )\pi$, $(m+(n+\frac{1}{2})\tau )\pi$ respectively.

27: 18.9 Recurrence Relations and Derivatives

… ►For $p_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)$, … ►For $p_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)$, … ►

Jacobi

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Jacobi

… ►Further $n$-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7). …

28: 20.3 Graphics

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Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

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Figure 20.3.3: ${\theta }_2(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

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Figure 20.3.4: ${\theta }_3(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

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Figure 20.3.5: ${\theta }_4(\pi x,q)$, $0\le x\le 2$, $q$ = 0. … Magnify

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Figure 20.3.18: ${\theta }_1\left(0.1\right|u+\mathrm{i}v)$, $-1\le u\le 1$, $0.005\le v\le 0.5$. …1 of $z$ is chosen arbitrarily since ${\theta }_1$ vanishes identically when $z=0$. Magnify 3D Help

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29: 20.9 Relations to Other Functions

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20.9.1 $$k={\theta }_2^2\left(0\right|\tau )/{\theta }_3^2\left(0\right|\tau )$$

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Defines:

$k$: modulus (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function and $\tau$: lattice parameter

Referenced by:

§20.11(iii), §20.9(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/20.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

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$K\left(k\right)=\frac{1}{2}\pi {\theta }_3^2\left(0\right|\tau ),$

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20.9.3 $$R_F(\frac{{\theta }_2^2(z,q)}{{\theta }_2^2(0,q)},\frac{{\theta }_3^2(z,q)}{{\theta }_3^2(0,q)},\frac{{\theta }_4^2(z,q)}{{\theta }_4^2(0,q)})=\frac{{\theta }_1^{\prime }(0,q)}{{\theta }_1(z,q)}z,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $z$: complex and $q$: nome

Referenced by:

§19.25(v), §20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E3

Encodings:

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See also:

Annotations for §20.9(i), §20.9 and Ch.20

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20.9.4 $$R_F(0,{\theta }_3^4(0,q),{\theta }_4^4(0,q))=\frac{1}{2}\pi ,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter and $q$: nome

Referenced by:

§20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E4

Encodings:

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See also:

Annotations for §20.9(i), §20.9 and Ch.20

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

30: 22.18 Mathematical Applications

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Ellipse

… ►where $ℰ(u,k)$ is Jacobi’s epsilon function (§22.16(ii)). … ►By use of the functions $\mathrm{sn}$ and $\mathrm{cn}$, parametrizations of algebraic equations, such as … ►

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

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