§19.25 Relations to Other Functions

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

Let ${k^{\prime}}^{2}=1-k^{2}$ and $c={\csc^{2}}\phi$. Then

 19.25.1 $\displaystyle K\left(k\right)$ $\displaystyle=R_{F}\left(0,{k^{\prime}}^{2},1\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=2\!R_{G}\left(0,{k^{\prime}}^{2},1\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=\tfrac{1}{3}{k^{\prime}}^{2}\left(R_{D}\left(0,{k^{\prime}}^{2},% 1\right)+R_{D}\left(0,1,{k^{\prime}}^{2}\right)\right),$ $\displaystyle K\left(k\right)-E\left(k\right)$ $\displaystyle=k^{2}D\left(k\right)=\tfrac{1}{3}k^{2}R_{D}\left(0,{k^{\prime}}^% {2},1\right),$ $\displaystyle E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)$ $\displaystyle=\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(0,1,{k^{\prime}}^{2}% \right).$
 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 $E\left(\phi,k\right)=2\!R_{G}\left(c-1,c-k^{2},c\right)-(c-1)R_{F}\left(c-1,c-% k^{2},c\right)-\sqrt{(c-1)(c-k^{2})/c},$
 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 $E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c-1,c-k^{2},c\right)+\tfrac{1}% {3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}\right)+k^{2}\sqrt{(c-1)/(c(c-% k^{2}))},$ $c>k^{2}$,
 19.25.11 $E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2}R_{D}\left(c-k^{2},c,c-1% \right)+\sqrt{(c-k^{2})/(c(c-1))},$ $\phi\neq\tfrac{1}{2}\pi$.

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

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

Assume $0\leq x\leq y\leq z$, $x, 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.

 19.25.35 $z=R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)% -e_{3}\right),$

provided that $z$ satisfies

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

and the left-hand side does not vanish for more than one value of $j$. Also,

 19.25.37 $\zeta\left(z\right)+z\wp\left(z\right)=2\!R_{G}\left(\wp\left(z\right)-e_{1},% \wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right).$ ⓘ 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), 6th Erratum Permalink: http://dlmf.nist.gov/19.25.E37 Encodings: TeX, pMML, png 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 (19.25.38) and (19.25.39) $j,k,\ell$ is any permutation of the numbers $1,2,3$.

 19.25.38 $\omega_{j}=R_{F}\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right),$
 19.25.39 $\eta_{j}+\omega_{j}e_{j}=2\!R_{G}\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right).$

For $\eta_{j}$ see (23.2.12).

Lastly,

 19.25.40 $z=\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z),\sigma_{3% }^{2}(z)\right),$

where

 19.25.41 $\sigma_{j}(z)=\exp\left(-\eta_{j}z\right)\sigma\left(z+\omega_{j}\right)/% \sigma\left(\omega_{j}\right),$ $j=1,2,3$. ⓘ 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

For $\eta_{j}$ see (23.2.12).

§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.)