Jacobi’s

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1: 22.16 Related Functions

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§22.16(ii) Jacobi’s Epsilon Function

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Integral Representations

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§22.16(iii) Jacobi’s Zeta Function

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Figure 22.16.2: Jacobi’s epsilon function $ℰ(x,k)$ for $0\le x\le 10\pi$ and $k=0.4,0.7,0.99,0.999999$. … Magnify

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Figure 22.16.3: Jacobi’s zeta function $\mathrm{Z}\left(x|k\right)$ for $0\le x\le 10\pi$ and $k=0.4,0.7,0.99,0.999999$. Magnify

2: 22.21 Tables

§22.21 Tables

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3: 20.4 Values at $z$ = 0

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Jacobi’s Identity

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4: 20.1 Special Notation

… ►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 $u=z/{\theta }_3^2\left(0\right|\tau )$. … ►McKean and Moll’s notation: ${\vartheta }_j(z|\tau )={\theta }_j\left(\pi z\right|\tau )$, $j=1,2,3,4$. …

5: 27.13 Functions

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27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

►One of Jacobi’s identities implies that … ►Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

6: 22.6 Elementary Identities

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22.6.22 $${\mathrm{p}\mathrm{q}}^2(\frac{1}{2}z,k)=\frac{\mathrm{p}\mathrm{s}(z,k)+\mathrm{r}\mathrm{s}(z,k)}{\mathrm{q}\mathrm{s}(z,k)+\mathrm{r}\mathrm{s}(z,k)}=\frac{\mathrm{p}\mathrm{q}(z,k)+\mathrm{r}\mathrm{q}(z,k)}{1+\mathrm{r}\mathrm{q}(z,k)}=\frac{\mathrm{p}\mathrm{r}(z,k)+1}{\mathrm{q}\mathrm{r}(z,k)+1}.$$

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Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function, $z$: complex and $k$: modulus

Referenced by:

§22.6(iii)

Permalink:

http://dlmf.nist.gov/22.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(iii), §22.6 and Ch.22

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

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

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$\mathrm{sn}(\mathrm{i}z,k)$ $=$	$\mathrm{i}\mathrm{sc}(z,k^{\prime })$	$\mathrm{dc}(\mathrm{i}z,k)$ $=$	$\mathrm{dn}(z,k^{\prime })$
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7: 25.1 Special Notation

… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

8: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. … ►Other notations for $\mathrm{sn}(z,k)$ are $\mathrm{sn}(z|m)$ and $\mathrm{sn}(z,m)$ with $m=k^2$; see Abramowitz and Stegun (1964) and Walker (1996). …

9: 20.9 Relations to Other Functions

… ►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). …

10: 19.5 Maclaurin and Related Expansions

… ► ►For Jacobi’s nome $q$: ►

19.5.5 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)=r+8r^2+84r^3+992r^4+\mathrm{\cdots },$$ $r=\frac{1}{16}{k}^2$, $0\le k\le 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function, $q$: nome and $k$: real or complex modulus

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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