# §19.25 Relations to Other Functions

## §19.25(i) Legendre’s Integrals as Symmetric Integrals

Let ${k^{\prime}}^{2}=1-k^{2}$ and $c={\csc}^{2}\phi$ with $0\leq\Re\phi\leq\ifrac{\pi}{2}$. Then

 19.25.1 $\displaystyle K\left(k\right)$ $\displaystyle=R_{F}\left(0,{k^{\prime}}^{2},1\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=2R_{G}\left(0,{k^{\prime}}^{2},1\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=\tfrac{1}{3}{k^{\prime}}^{2}\left(R_{D}\left(0,{k^{\prime}}^{2},% 1\right)+R_{D}\left(0,1,{k^{\prime}}^{2}\right)\right),$ $\displaystyle K\left(k\right)-E\left(k\right)$ $\displaystyle=k^{2}D\left(k\right)=\tfrac{1}{3}k^{2}R_{D}\left(0,{k^{\prime}}^% {2},1\right),$ $\displaystyle E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)$ $\displaystyle=\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(0,1,{k^{\prime}}^{2}% \right).$
 19.25.2 $\Pi\left(\alpha^{2},k\right)-K\left(k\right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(% 0,{k^{\prime}}^{2},1,1-\alpha^{2}\right).$
 19.25.3 $\Pi\left(\alpha^{2},k\right)=\tfrac{1}{2}\pi R_{-\frac{1}{2}}\left(\tfrac{1}{2% },-\tfrac{1}{2},1;{k^{\prime}}^{2},1,1-\alpha^{2}\right),$

with Cauchy principal value

 19.25.4 $\Pi\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/\alpha^{2})R_{J}\left(0,1-k^{% 2},1,1-(k^{2}/\alpha^{2})\right),$ $-\infty.
 19.25.5 $\displaystyle F\left(\phi,k\right)$ $\displaystyle=R_{F}\left(c-1,c-k^{2},c\right),$ 19.25.6 $\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}$ $\displaystyle=\tfrac{1}{3}kR_{D}\left(c-1,c,c-k^{2}\right).$
 19.25.7 $E\left(\phi,k\right)=2R_{G}\left(c-1,c-k^{2},c\right)-(c-1)R_{F}\left(c-1,c-k^% {2},c\right)-\ifrac{\sqrt{c-1}\sqrt{c-k^{2}}}{\sqrt{c}},$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind, $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $\phi$: real or complex argument and $k$: real or complex modulus Referenced by: §19.25(i), §19.25(i), §19.25(iii), Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.36(ii), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.25.E7 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior. Suggested 2021-06-07 by Luc Maisonobe See also: Annotations for §19.25(i), §19.25 and Ch.19
 19.25.8 $E\left(\phi,k\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{3% }{2};c-1,c-k^{2},c\right),$
 19.25.9 $E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}% \left(c-1,c-k^{2},c\right),$
 19.25.10 $E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c-1,c-k^{2},c\right)+\tfrac{1}% {3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}\right)+k^{2}\ifrac{\sqrt{c-1}% }{\left(\sqrt{c}\sqrt{c-k^{2}}\right)},$ $c>k^{2}$, ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $\phi$: real or complex argument, $k$: real or complex modulus and $k^{\prime}$: complementary modulus Referenced by: §19.25(i), §19.30(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.25.E10 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior. Suggested 2021-06-07 by Luc Maisonobe See also: Annotations for §19.25(i), §19.25 and Ch.19
 19.25.11 $E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2}R_{D}\left(c-k^{2},c,c-1% \right)+\ifrac{\sqrt{c-k^{2}}}{\left(\sqrt{c}\sqrt{c-1}\right)},$ $\phi\neq\tfrac{1}{2}\pi$. ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $\phi$: real or complex argument, $k$: real or complex modulus and $k^{\prime}$: complementary modulus Referenced by: §19.25(i), §19.25(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.25.E11 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior. Suggested 2021-06-07 by Luc Maisonobe See also: Annotations for §19.25(i), §19.25 and Ch.19

Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of $R_{D}$; see (19.21.7). All terms on the right-hand sides are nonnegative when $k^{2}\leq 0$, $0\leq k^{2}\leq 1$, or $1\leq k^{2}\leq c$, respectively.

 19.25.12 $\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,% c-k^{2},c\right).$
 19.25.13 $D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right).$
 19.25.14 $\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right)=\tfrac{1}{3}\alpha^{2}R% _{J}\left(c-1,c-k^{2},c,c-\alpha^{2}\right),$
 19.25.15 $\Pi\left(\phi,\alpha^{2},k\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},\tfrac{1}% {2},-\tfrac{1}{2},1;c-1,c-k^{2},c,c-\alpha^{2}\right).$

If $\alpha^{2}>c$, then the Cauchy principal value is

 19.25.16 $\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2% },c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2% })}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})% \right),$ $\omega^{2}=k^{2}/\alpha^{2}$.

The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). For example, if we write (19.25.5) as

 19.25.17 $F\left(\phi,k\right)=R_{F}\left(x,y,z\right),$

with

 19.25.18 $(x,y,z)=(c-1,c-k^{2},c),$ ⓘ Symbols: $k$: real or complex modulus Permalink: http://dlmf.nist.gov/19.25.E18 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

then the five nontrivial permutations of $x,y,z$ that leave $R_{F}$ invariant change $k^{2}$ ($=(z-y)/(z-x)$) into $1/k^{2}$, ${k^{\prime}}^{2}$, $1/{k^{\prime}}^{2}$, $-k^{2}/{k^{\prime}}^{2}$, $-{k^{\prime}}^{2}/k^{2}$, and $\sin\phi$ ($=\sqrt{(z-x)/z}$) into $k\sin\phi$, $-i\tan\phi$, $-ik^{\prime}\tan\phi$, $(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin}^{2}\phi}$, $-ik\sin\phi/\sqrt{1-k^{2}{\sin}^{2}\phi}$. Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions).

The three changes of parameter of $\Pi\left(\phi,\alpha^{2},k\right)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14).

## §19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

Let $r=1/x^{2}$. Then

 19.25.19 $\displaystyle\operatorname{cel}\left(k_{c},p,a,b\right)$ $\displaystyle=aR_{F}\left(0,k_{c}^{2},1\right)+\tfrac{1}{3}{(b-pa)}R_{J}\left(% 0,k_{c}^{2},1,p\right),$ 19.25.20 $\displaystyle\operatorname{el1}\left(x,k_{c}\right)$ $\displaystyle=R_{F}\left(r,r+k_{c}^{2},r+1\right),$ 19.25.21 $\displaystyle\operatorname{el2}\left(x,k_{c},a,b\right)$ $\displaystyle=a\operatorname{el1}\left(x,k_{c}\right)+\tfrac{1}{3}{(b-a)}R_{D}% \left(r,r+k_{c}^{2},r+1\right),$ 19.25.22 $\displaystyle\operatorname{el3}\left(x,k_{c},p\right)$ $\displaystyle=\operatorname{el1}\left(x,k_{c}\right)+\tfrac{1}{3}{(1-p)}R_{J}% \left(r,r+k_{c}^{2},r+1,r+p\right).$

## §19.25(iii) Symmetric Integrals as Legendre’s Integrals

Assume $0\leq x\leq y\leq z$, $x, $(x,y)\neq(0,0)$ and $p>0$. Let

 19.25.23 $\displaystyle\phi$ $\displaystyle=\operatorname{arccos}\sqrt{\ifrac{x}{z}}=\operatorname{arcsin}% \sqrt{\ifrac{(z-x)}{z}},$ $\displaystyle k$ $\displaystyle=\sqrt{\frac{z-y}{z-x}},$ $\displaystyle\alpha^{2}$ $\displaystyle=\frac{z-p}{z-x},$

with $\alpha\neq 0$. Then

 19.25.24 $(z-x)^{1/2}R_{F}\left(x,y,z\right)=F\left(\phi,k\right),$
 19.25.25 $(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F\left(\phi,k\right)-E\left(\phi,% k\right)),$
 19.25.26 $(z-x)^{3/2}R_{J}\left(x,y,z,p\right)=(3/\alpha^{2}){(\Pi\left(\phi,\alpha^{2},% k\right)-F\left(\phi,k\right))},$
 19.25.27 $2(z-x)^{-1/2}R_{G}\left(x,y,z\right)=E\left(\phi,k\right)+(\cot\phi)^{2}F\left% (\phi,k\right)+(\cot\phi)\sqrt{1-k^{2}{\sin}^{2}\phi}.$

## §19.25(iv) Theta Functions

For relations of symmetric integrals to theta functions, see §20.9(i).

## §19.25(v) Jacobian Elliptic Functions

For the notation see §§22.2, 22.15, and 22.16(i).

With $0\leq k^{2}\leq 1$ and $\mathrm{p,q,r}$ any permutation of the letters $\mathrm{c,d,n}$, define

 19.25.28 $\Delta(\mathrm{p,q})={\operatorname{ps}}^{2}\left(u,k\right)-{\operatorname{qs% }}^{2}\left(u,k\right)=-\Delta(\mathrm{q,p}),$ ⓘ Defines: $\Delta(\mathrm{p,q})$: function (locally) Symbols: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$: generic Jacobian elliptic function and $k$: real or complex modulus Permalink: http://dlmf.nist.gov/19.25.E28 Encodings: TeX, pMML, png See also: Annotations for §19.25(v), §19.25 and Ch.19

which implies

 19.25.29 $\displaystyle\Delta(\mathrm{n,d})$ $\displaystyle=k^{2},$ $\displaystyle\Delta(\mathrm{d,c})$ $\displaystyle={k^{\prime}}^{2},$ $\displaystyle\Delta(\mathrm{n,c})$ $\displaystyle=1.$ ⓘ Symbols: $k$: real or complex modulus, $k^{\prime}$: complementary modulus and $\Delta(\mathrm{p,q})$: function Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/19.25.E29 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.25(v), §19.25 and Ch.19

If ${\operatorname{cs}}^{2}\left(u,k\right)\geq 0$, then

 19.25.30 $\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),$
 19.25.31 $u=R_{F}\left({\operatorname{ps}}^{2}\left(u,k\right),{\operatorname{qs}}^{2}% \left(u,k\right),{\operatorname{rs}}^{2}\left(u,k\right)\right);$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$: generic Jacobian elliptic function and $k$: real or complex modulus Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/19.25.E31 Encodings: TeX, pMML, png See also: Annotations for §19.25(v), §19.25 and Ch.19

compare (19.25.35) and (20.9.3).

 19.25.32 $\operatorname{arcps}\left(x,k\right)=R_{F}\left(x^{2},x^{2}+\Delta(\mathrm{q,p% }),x^{2}+\Delta(\mathrm{r,p})\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $k$: real or complex modulus and $\Delta(\mathrm{p,q})$: function Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/19.25.E32 Encodings: TeX, pMML, png See also: Annotations for §19.25(v), §19.25 and Ch.19
 19.25.33 $\operatorname{arcsp}\left(x,k\right)=xR_{F}\left(1,1+\Delta(\mathrm{q,p})x^{2}% ,1+\Delta(\mathrm{r,p})x^{2}\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $k$: real or complex modulus and $\Delta(\mathrm{p,q})$: function Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/19.25.E33 Encodings: TeX, pMML, png See also: Annotations for §19.25(v), §19.25 and Ch.19
 19.25.34 $\operatorname{arcpq}\left(x,k\right)=\sqrt{w}R_{F}\left(x^{2},1,1+\Delta(% \mathrm{r,q})w\right),$ $w=\ifrac{(1-x^{2})}{\Delta(\mathrm{q,p})}$, ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $k$: real or complex modulus and $\Delta(\mathrm{p,q})$: function Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/19.25.E34 Encodings: TeX, pMML, png See also: Annotations for §19.25(v), §19.25 and Ch.19

where we assume $0\leq x^{2}\leq 1$ if $x=\operatorname{sn}$, $\operatorname{cn}$, or $\operatorname{cd}$; $x^{2}\geq 1$ if $x=\operatorname{ns}$, $\operatorname{nc}$, or $\operatorname{dc}$; $x$ real if $x=\operatorname{cs}$ or $\operatorname{sc}$; $k^{\prime}\leq x\leq 1$ if $x=\operatorname{dn}$; $1\leq x\leq 1/k^{\prime}$ if $x=\operatorname{nd}$; $x^{2}\geq{k^{\prime}}^{2}$ if $x=\operatorname{ds}$; $0\leq x^{2}\leq 1/{k^{\prime}}^{2}$ if $x=\operatorname{sd}$.

For the use of $R$-functions with $\Delta(\mathrm{p,q})$ in unifying other properties of Jacobian elliptic functions, see Carlson (2004, 2006a, 2006b, 2008).

Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_{F}\left(x,y,z\right)$. See (19.29.19), Carlson (2005), and (22.15.11), and compare with Abramowitz and Stegun (1964, (17.4.41)–(17.4.52)). For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9).

## §19.25(vi) Weierstrass Elliptic Functions

For the notation see §23.2 and §23.3. Let $\mathbb{L}$ be a lattice for the Weierstrass elliptic function $\wp\left(z\right)$. Then

 19.25.35 $z+2\omega=\pm R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp% \left(z\right)-e_{3}\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$): Weierstrass $\wp$-function, $e_{\NVar{j}}$: Weierstrass lattice roots and $z$: complex number Proof sketch: Use (23.6.36) with $z=\wp\left(\omega\right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\wp\left(\omega\right)$ and compare with (19.16.1). The undetermined $\pm$ is a consequence of the multivaluedness of the square-roots in (19.16.4). Referenced by: (19.25.38), (19.25.40), §19.25(v), §19.25(vi), §19.25(vi), §20.9(i), Erratum (V1.1.0) for Subsection 19.25(vi) Permalink: http://dlmf.nist.gov/19.25.E35 Encodings: TeX, pMML, png Correction (effective with 1.1.0): This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$. See also: Annotations for §19.25(vi), §19.25 and Ch.19

for some $2\omega\in\mathbb{L}$, provided that $z$ satisfies

 19.25.36 $\wp\left(z\right)-e_{j}\in\mathbb{C}\setminus(-\infty,0],$ $j=1,2,3.$

The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which $\wp\left(z\right)-e_{j}<0$, for some $j$. Also,

 19.25.37 $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)=\pm 2R_{G}\left(\wp% \left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right),$ ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$): Weierstrass $\wp$-function, $e_{\NVar{j}}$: Weierstrass lattice roots, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$): Weierstrass zeta function and $z$: complex number Referenced by: §19.25(vi), Erratum (V1.0.18) for Equation (19.25.37), Erratum (V1.1.0) for Subsection 19.25(vi) Permalink: http://dlmf.nist.gov/19.25.E37 Encodings: TeX, pMML, png Correction (effective with 1.1.0): This equation has been corrected, namely the left-hand side which was originally $\zeta\left(z\right)+z\wp\left(z\right)$, has been replaced by $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)$, and the right-hand side has been multiplied by $\pm 1$. Clarification (effective with 1.0.18): The Weierstrass zeta function in this equation incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the functions appeared correct. See also: Annotations for §19.25(vi), §19.25 and Ch.19

in which the sign and the $\omega$ are the same as in (19.25.35).

In (19.25.38) and (19.25.39) $j$, $k$, $\ell$ is any permutation of the numbers $1,2,3$.

 19.25.38 $\omega_{j}=R_{F}\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $e_{\NVar{j}}$: Weierstrass lattice roots, $j$: $(=1,2,3)$, $z$: complex number, $k$: $(=1,2,3)$, $\ell$: $(=1,2,3)$ and $\omega_{j}$: nonzero complex number Proof sketch: Take $z=-\omega$ in (19.25.35). Referenced by: §19.25(vi), §19.25(vi) Permalink: http://dlmf.nist.gov/19.25.E38 Encodings: TeX, pMML, png See also: Annotations for §19.25(vi), §19.25 and Ch.19
 19.25.39 $\zeta\left(\omega_{j}\right)+\omega_{j}e_{j}=2R_{G}\left(0,e_{j}-e_{k},e_{j}-e% _{\ell}\right),$ ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind, $e_{\NVar{j}}$: Weierstrass lattice roots, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$): Weierstrass zeta function, $j$: $(=1,2,3)$, $z$: complex number, $k$: $(=1,2,3)$, $\ell$: $(=1,2,3)$, $\omega_{j}$: nonzero complex number and $\eta_{j}$: complex number Proof sketch: Take $z=-\omega$ in (19.25.36). Referenced by: §19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi) Permalink: http://dlmf.nist.gov/19.25.E39 Encodings: TeX, pMML, png Correction (effective with 1.1.0): This equation has been corrected such that the left-hand side $\eta_{j}$ has been replaced by $\zeta\left(\omega_{j}\right)$. See also: Annotations for §19.25(vi), §19.25 and Ch.19

for some $2\omega_{j}\in\mathbb{L}$ and $\wp\left(\omega_{j}\right)=e_{j}$.

Lastly,

 19.25.40 $z+2\omega=\pm\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z% ),\sigma_{3}^{2}(z)\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$): Weierstrass sigma function, $z$: complex number and $\sigma_{j}(z)$: function Proof sketch: Combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)), (19.25.35) and (19.16.1). Referenced by: §19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi) Permalink: http://dlmf.nist.gov/19.25.E40 Encodings: TeX, pMML, png Correction (effective with 1.1.0): This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$. See also: Annotations for §19.25(vi), §19.25 and Ch.19

for some $2\omega\in\mathbb{L}$, where

 19.25.41 $\sigma_{j}(z)=\exp\left(-\eta_{j}z\right)\ifrac{\sigma\left(z+\omega_{j}\right% )}{\sigma\left(\omega_{j}\right)},$ $j=1,2,3,$ ⓘ Defines: $\sigma_{j}(z)$: function (locally) Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$): Weierstrass sigma function, $\exp\NVar{z}$: exponential function, $j$: $(=1,2,3)$, $z$: complex number, $\omega_{j}$: nonzero complex number and $\eta_{j}$: complex number Permalink: http://dlmf.nist.gov/19.25.E41 Encodings: TeX, pMML, png See also: Annotations for §19.25(vi), §19.25 and Ch.19

in which $2\omega_{1}$ and $2\omega_{3}$ are generators for the lattice $\mathbb{L}$, $\omega_{2}=-\omega_{1}-\omega_{3}$, and $\eta_{j}=\zeta\left(\omega_{j}\right)$ (see (23.2.12)). The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which $\sigma_{j}^{2}(z)<0$, for some $j$.

## §19.25(vii) Hypergeometric Function

 19.25.42 ${{}_{2}F_{1}}\left(a,b;c;z\right)=R_{-a}\left(b,c-b;1-z,1\right),$
 19.25.43 $R_{-a}\left(b_{1},b_{2};z_{1},z_{2}\right)=z_{2}^{-a}{{}_{2}F_{1}}\left(a,b_{1% };b_{1}+b_{2};1-(z_{1}/z_{2})\right).$

For these results and extensions to the Appell function ${F_{1}}$16.13) and Lauricella’s function $F_{D}$ see Carlson (1963). (${F_{1}}$ and $F_{D}$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.)