# §23.6 Relations to Other Functions

## §23.6(i) Theta Functions

In this subsection $2\omega_{1}$, $2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$, and the lattice roots $e_{1}$, $e_{2}$, $e_{3}$ are given by (23.3.9).

 23.6.1 $\displaystyle q$ $\displaystyle=e^{i\pi\tau},$ $\displaystyle\tau$ $\displaystyle=\omega_{3}/\omega_{1}.$
 23.6.2 $\displaystyle e_{1}$ $\displaystyle=\frac{\pi^{2}}{12\omega_{1}^{2}}\left({\theta_{2}}^{4}\left(0,q% \right)+2{\theta_{4}}^{4}\left(0,q\right)\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $e_{\NVar{j}}$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathbb{L}$: lattice, $q$: nome and $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators A&S Ref: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11 Referenced by: §23.22(ii), §23.22(i), §23.6(i), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E2 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23 23.6.3 $\displaystyle e_{2}$ $\displaystyle=\frac{\pi^{2}}{12\omega_{1}^{2}}\left({\theta_{2}}^{4}\left(0,q% \right)-{\theta_{4}}^{4}\left(0,q\right)\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $e_{\NVar{j}}$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathbb{L}$: lattice, $q$: nome and $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators A&S Ref: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11 Permalink: http://dlmf.nist.gov/23.6.E3 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23 23.6.4 $\displaystyle e_{3}$ $\displaystyle=-\frac{\pi^{2}}{12\omega_{1}^{2}}\left(2{\theta_{2}}^{4}\left(0,% q\right)+{\theta_{4}}^{4}\left(0,q\right)\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $e_{\NVar{j}}$: Weierstrass lattice roots, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathbb{L}$: lattice, $q$: nome and $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators A&S Ref: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11 Referenced by: §23.22(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E4 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23
 23.6.5 $\displaystyle\wp\left(z\right)-e_{1}$ $\displaystyle=\left(\frac{\pi\theta_{3}\left(0,q\right)\theta_{4}\left(0,q% \right)\theta_{2}\left(\pi z/(2\omega_{1}),q\right)}{2\omega_{1}\theta_{1}% \left(\pi z/(2\omega_{1}),q\right)}\right)^{2},$ 23.6.6 $\displaystyle\wp\left(z\right)-e_{2}$ $\displaystyle=\left(\frac{\pi\theta_{2}\left(0,q\right)\theta_{4}\left(0,q% \right)\theta_{3}\left(\pi z/(2\omega_{1}),q\right)}{2\omega_{1}\theta_{1}% \left(\pi z/(2\omega_{1}),q\right)}\right)^{2},$ 23.6.7 $\displaystyle\wp\left(z\right)-e_{3}$ $\displaystyle=\left(\frac{\pi\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(\pi z/(2\omega_{1}),q\right)}{2\omega_{1}\theta_{1}% \left(\pi z/(2\omega_{1}),q\right)}\right)^{2}.$
 23.6.8 $\eta_{1}=-\frac{\pi^{2}}{12\omega_{1}}\frac{\theta_{1}'''\left(0,q\right)}{% \theta_{1}'\left(0,q\right)}.$
 23.6.9 $\sigma\left(z\right)=2\omega_{1}\exp\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}% \right)\frac{\theta_{1}\left(\pi z/(2\omega_{1}),q\right)}{\pi\theta_{1}'\left% (0,q\right)},$
 23.6.10 $\displaystyle\sigma\left(\omega_{1}\right)$ $\displaystyle=2\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\right)% \theta_{2}\left(0,q\right)}{\pi\theta_{1}'\left(0,q\right)},$ 23.6.11 $\displaystyle\sigma\left(\omega_{2}\right)$ $\displaystyle=-2\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}(\omega_{1}\tau^% {2}+\omega_{3}-\omega_{2})\right)\theta_{3}\left(0,q\right)}{\pi q^{1/4}\theta% _{1}'\left(0,q\right)},$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{L}$: lattice, $q$: nome, $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators, $\tau$: complex variable and $\eta_{j}$: complex number Referenced by: Erratum (V1.1.1) for Equation (23.6.11) Permalink: http://dlmf.nist.gov/23.6.E11 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The factor $2\omega_{1}i\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\tau^{2}\right)$ has been corrected to be $-2\omega_{1}\exp\left(\tfrac{1}{2}\eta_{1}(\omega_{1}\tau^{2}+\omega_{3}-% \omega_{2})\right)$. See also: Annotations for §23.6(i), §23.6 and Ch.23 23.6.12 $\displaystyle\sigma\left(\omega_{3}\right)$ $\displaystyle=2i\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\tau^{% 2}\right)\theta_{4}\left(0,q\right)}{\pi q^{1/4}\theta_{1}'\left(0,q\right)}.$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{L}$: lattice, $q$: nome, $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators, $\tau$: complex variable and $\eta_{j}$: complex number Referenced by: §23.6(i), Erratum (V1.1.1) for Equation (23.6.12) Permalink: http://dlmf.nist.gov/23.6.E12 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The factor $-2\omega_{1}\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\right)$ has been corrected to be $2i\omega_{1}\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\tau^{2}\right)$. See also: Annotations for §23.6(i), §23.6 and Ch.23

With $z=\ifrac{\pi u}{(2\omega_{1})}$,

 23.6.14 $\wp\left(u\right)=\left(\frac{\pi}{2\omega_{1}}\right)^{2}\left(\frac{\theta_{% 1}'''\left(0,q\right)}{3\theta_{1}'\left(0,q\right)}-\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}z}^{2}}\ln\theta_{1}\left(z,q\right)\right),$
 23.6.15 $\frac{\sigma\left(u+\omega_{j}\right)}{\sigma\left(\omega_{j}\right)}=\exp% \left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)\frac{\theta_{j+1}% \left(z,q\right)}{\theta_{j+1}\left(0,q\right)},$ $j=1,2,3$. ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$): Weierstrass sigma function, $\exp\NVar{z}$: exponential function, $\mathbb{L}$: lattice, $q$: nome, $z$: complex, $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators, $\eta_{j}$: complex number and $u$: change of variable Referenced by: §23.6(i), Erratum (V1.1.1) for Equation (23.6.15) Permalink: http://dlmf.nist.gov/23.6.E15 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The factor $\exp\left(\eta_{j}u+\frac{\eta_{j}u^{2}}{2\omega_{1}}\right)$ has been corrected to be $\exp\left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)$. Suggested 2021-02-10 by Jan Felipe van Diejen See also: Annotations for §23.6(i), §23.6 and Ch.23

For further results for the $\sigma$-function see Lawden (1989, §6.2).

## §23.6(ii) Jacobian Elliptic Functions

Again, in Equations (23.6.16)–(23.6.26), $2\omega_{1},2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$ and $e_{1},e_{2},e_{3}$ are given by (23.3.9).

 23.6.16 $\displaystyle k^{2}$ $\displaystyle=\frac{e_{2}-e_{3}}{e_{1}-e_{3}},$ $\displaystyle{k^{\prime}}^{2}$ $\displaystyle=\frac{e_{1}-e_{2}}{e_{1}-e_{3}},$ ⓘ Symbols: $e_{\NVar{j}}$: Weierstrass lattice roots, $\mathbb{L}$: lattice, $k$: modulus and $k^{\prime}$: complementary modulus A&S Ref: 18.9.9 Referenced by: (23.22.1), item 1, §23.22(ii), §23.6(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
 23.6.17 $\displaystyle{K}^{2}$ $\displaystyle=(K\left(k\right))^{2}=\omega_{1}^{2}(e_{1}-e_{3}),$ $\displaystyle{{K^{\prime}}}^{2}$ $\displaystyle=(K\left(k^{\prime}\right))^{2}=\omega_{3}^{2}(e_{3}-e_{1}).$
 23.6.18 $\displaystyle e_{1}$ $\displaystyle=\frac{{K}^{2}}{3\omega_{1}^{2}}(1+{k^{\prime}}^{2}),$ 23.6.19 $\displaystyle e_{2}$ $\displaystyle=\frac{{K}^{2}}{3\omega_{1}^{2}}(k^{2}-{k^{\prime}}^{2}),$ 23.6.20 $\displaystyle e_{3}$ $\displaystyle=-\frac{{K}^{2}}{3\omega_{1}^{2}}(1+k^{2}).$
 23.6.21 $\displaystyle\wp\left(z\right)-e_{1}$ $\displaystyle=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{cs}}^{2}\left(\frac% {K\!z}{\omega_{1}},k\right),$ 23.6.22 $\displaystyle\wp\left(z\right)-e_{2}$ $\displaystyle=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{ds}}^{2}\left(\frac% {K\!z}{\omega_{1}},k\right),$ 23.6.23 $\displaystyle\wp\left(z\right)-e_{3}$ $\displaystyle=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{ns}}^{2}\left(\frac% {K\!z}{\omega_{1}},k\right).$
 23.6.24 $\displaystyle\wp\left(z+\omega_{1}\right)-e_{1}$ $\displaystyle=\left(\frac{Kk^{\prime}}{\omega_{1}}\right)^{2}{\operatorname{sc% }}^{2}\left(\frac{K\!z}{\omega_{1}},k\right),$ 23.6.25 $\displaystyle\wp\left(z+\omega_{2}\right)-e_{2}$ $\displaystyle=-\left(\frac{Kkk^{\prime}}{\omega_{1}}\right)^{2}{\operatorname{% sd}}^{2}\left(\frac{K\!z}{\omega_{1}},k\right),$ 23.6.26 $\displaystyle\wp\left(z+\omega_{3}\right)-e_{3}$ $\displaystyle=\left(\frac{Kk}{\omega_{1}}\right)^{2}{\operatorname{sn}}^{2}% \left(\frac{K\!z}{\omega_{1}},k\right).$

In (23.6.27)–(23.6.29) the modulus $k$ is given and $K=K\left(k\right)$, ${K^{\prime}}=K\left(k^{\prime}\right)$ are the corresponding complete elliptic integrals (§19.2(ii)). Also, $\mathbb{L}_{\mspace{1.0mu}1}$, $\mathbb{L}_{\mspace{1.0mu}2}$, $\mathbb{L}_{\mspace{1.0mu}3}$ are the lattices with generators $(4K,2\mathrm{i}{K^{\prime}})$, $(2K-2\mathrm{i}{K^{\prime}},2K+2\mathrm{i}{K^{\prime}})$, $(2K,4\mathrm{i}{K^{\prime}})$, respectively.

 23.6.27 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu}1}\right)-\zeta\left(z+2K|\mathbb{L}_{% \mspace{1.0mu}1}\right)+\zeta\left(2K|\mathbb{L}_{\mspace{1.0mu}1}\right)=% \operatorname{ns}\left(z,k\right),$
 23.6.28 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu}2}\right)-\zeta\left(z+2K|\mathbb{L}_{% \mspace{1.0mu}2}\right)+\zeta\left(2K|\mathbb{L}_{\mspace{1.0mu}2}\right)=% \operatorname{ds}\left(z,k\right),$
 23.6.29 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu}3}\right)-\zeta\left(z+2\mathrm{i}{K^{% \prime}}|\mathbb{L}_{\mspace{1.0mu}3}\right)-\zeta\left(2\mathrm{i}{K^{\prime}% }|\mathbb{L}_{\mspace{1.0mu}3}\right)=\operatorname{cs}\left(z,k\right).$

Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3.

For representations of the Jacobi functions $\operatorname{sn}$, $\operatorname{cn}$, and $\operatorname{dn}$ as quotients of $\sigma$-functions see Lawden (1989, §§6.2, 6.3).

## §23.6(iii) General Elliptic Functions

For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma\left(z\right)$ and $\wp\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta\left(z\right)$ see Lawden (1989, §8.11).

## §23.6(iv) Elliptic Integrals

### Rectangular Lattice

Let $z$ be on the perimeter of the rectangle with vertices $0,2\omega_{1},2\omega_{1}+2\omega_{3},2\omega_{3}$. Then $t=\wp\left(z\right)$ is real (§§23.5(i)23.5(ii)), and

 23.6.30 $z=\frac{1}{2}\int_{t}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{(u-e_{1})(u-e_{2})(u-% e_{3})}},$ $t\geq e_{1}$, $z\in(0,\omega_{1}]$,
 23.6.31 $\displaystyle z-\omega_{1}$ $\displaystyle=\frac{i}{2}\int_{t}^{e_{1}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(% u-e_{2})(u-e_{3})}},$ $e_{2}\leq t\leq e_{1}$, $z\in[\omega_{1},\omega_{1}+\omega_{3}]$, 23.6.32 $\displaystyle z-\omega_{3}$ $\displaystyle=\frac{1}{2}\int_{e_{3}}^{t}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(% e_{2}-u)(u-e_{3})}},$ $e_{3}\leq t\leq e_{2}$, $z\in[\omega_{3},\omega_{1}+\omega_{3}]$,
 23.6.33 $z=\frac{i}{2}\int_{-\infty}^{t}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(e_{2}-u)(e% _{3}-u)}},$ $t\leq e_{3}$, $z\in(0,\omega_{3}]$.
 23.6.34 $2\omega_{1}=\int_{e_{1}}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{(u-e_{1})(u-e_{2})% (u-e_{3})}}=\int_{e_{3}}^{e_{2}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(e_{2}-u)(% u-e_{3})}},$
 23.6.35 $2\omega_{3}=i\int_{e_{2}}^{e_{1}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(u-e_{2})% (u-e_{3})}}=i\int_{-\infty}^{e_{3}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(e_{2}-% u)(e_{3}-u)}}.$

For (23.6.30)–(23.6.35) and further identities see Lawden (1989, §6.12).

Let $z$ be a point of $\mathbb{C}$ different from $e_{1},e_{2},e_{3}$, and define $w$ by
 23.6.36 $w=\int_{z}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{4u^{3}-g_{2}u-g_{3}}}=\frac{1}{2% }\int_{z}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{(u-e_{1})(u-e_{2})(u-e_{3})}},$
where the integral is taken along any path from $z$ to $\infty$ that does not pass through any of $e_{1},e_{2},e_{3}$. Then $z=\wp\left(w\right)$, where the value of $w$ depends on the choice of path and determination of the square root; see McKean and Moll (1999, pp. 87–88 and §2.5).