19 Elliptic Integrals Symmetric Integrals 19.24 Inequalities 19.26 Addition Theorems

§19.25 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/19.25
See also:: Annotations for Ch.19

§19.25(i) Legendre’s Integrals as Symmetric Integrals
§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
§19.25(iii) Symmetric Integrals as Legendre’s Integrals
§19.25(iv) Theta Functions
§19.25(v) Jacobian Elliptic Functions
§19.25(vi) Weierstrass Elliptic Functions
§19.25(vii) Hypergeometric Function

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

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Keywords:: Legendre’s elliptic integrals, Legendre’s elliptic integrals, relations to other functions, symmetric elliptic integrals, symmetric elliptic integrals, transformations replaced by symmetry
Notes:: (19.25.1), (19.25.2), and (19.25.3) are derived from the incomplete cases. For (19.25.4) put $c=1$ in (19.25.16). (19.25.5) and (19.25.7) come from Carlson (1977b, (9.3-2) and (9.3-3)). For (19.25.6) and (19.25.12) apply (19.18.4) to (19.25.5) and (19.25.8), respectively. (19.25.8) and (19.25.15) are special cases of (19.16.12). To get (19.25.9), (19.25.10), and (19.25.11), let $(c-1,c-k^{2},c)=(x,y,z)$ and eliminate $R_{G}$ between (19.25.7) and each of the three forms of (19.25.10) obtained by permuting $x, y$ and $z$ . For (19.25.13) combine (19.2.6) and (19.25.9). For (19.25.14) see Zill and Carlson (1970, (2.5)). For (19.25.16) substitute (19.25.14) in (19.7.8) and use (19.2.20).
Referenced by:: §19.25(ii), §19.36(i), §19.7(ii), Erratum (V1.1.5) for §19.25(i), Erratum (V1.1.7) for Subsection 19.25(i)
Permalink:: http://dlmf.nist.gov/19.25.i
Correction (effective with 1.1.7):: The constraint $-\pi<\Re\phi\leq\pi$ was corrected to read $0\leq\Re\phi\leq\ifrac{\pi}{2}$ .
Suggested 2022-09-19 by Charles Karney
Addition (effective with 1.1.5):: The constraint $-\pi<\Re\phi\leq\pi$ was added just above (19.25.1).
See also:: Annotations for §19.25 and Ch.19

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	$\displaystyle K\left(k\right)$	$\displaystyle=R_{F}\left(0,{k^{\prime}}^{2},1\right),$
	$\displaystyle E\left(k\right)$	$\displaystyle=2R_{G}\left(0,{k^{\prime}}^{2},1\right),$
	$\displaystyle E\left(k\right)$	$\displaystyle=\tfrac{1}{3}{k^{\prime}}^{2}\left(R_{D}\left(0,{k^{\prime}}^{2},% 1\right)+R_{D}\left(0,1,{k^{\prime}}^{2}\right)\right),$
	$\displaystyle K\left(k\right)-E\left(k\right)$	$\displaystyle=k^{2}D\left(k\right)=\tfrac{1}{3}k^{2}R_{D}\left(0,{k^{\prime}}^% {2},1\right),$
	$\displaystyle E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)$	$\displaystyle=\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(0,1,{k^{\prime}}^{2}% \right).$
ⓘ 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, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $D\left(\NVar{k}\right)$ : complete elliptic integral of Legendre’s type, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.17, §19.17, §19.25(i), Figure 19.3.3, Figure 19.3.3, Figure 19.3.3, Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.30(i), §19.6(i), §19.7(i), §19.9(i), §20.9(i), Erratum (V1.1.5) for §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E1 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.2		$\Pi\left(\alpha^{2},k\right)-K\left(k\right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(% 0,{k^{\prime}}^{2},1,1-\alpha^{2}\right).$
ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.12, §19.25(i), §19.6(i) Permalink: http://dlmf.nist.gov/19.25.E2 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.3		$\Pi\left(\alpha^{2},k\right)=\tfrac{1}{2}\pi R_{-\frac{1}{2}}\left(\tfrac{1}{2% },-\tfrac{1}{2},1;{k^{\prime}}^{2},1,1-\alpha^{2}\right),$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E3 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

with Cauchy principal value

19.25.4		$\Pi\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/\alpha^{2})R_{J}\left(0,1-k^{% 2},1,1-(k^{2}/\alpha^{2})\right),$
$-\infty<k^{2}<1<\alpha^{2}$ .
ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E4 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.5	$\displaystyle F\left(\phi,k\right)$	$\displaystyle=R_{F}\left(c-1,c-k^{2},c\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.25(i), §19.25(i), §19.25(iii), Figure 19.3.3, Figure 19.3.3, Figure 19.3.3, §19.36(ii), §19.6(ii), §19.9(ii), §19.9(ii) Permalink: http://dlmf.nist.gov/19.25.E5 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19
19.25.6	$\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}$	$\displaystyle=\tfrac{1}{3}kR_{D}\left(c-1,c,c-k^{2}\right).$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E6 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

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		$E\left(\phi,k\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{3% }{2};c-1,c-k^{2},c\right),$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $E\left(\NVar{\phi},\NVar{k}\right)$ : 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) Permalink: http://dlmf.nist.gov/19.25.E8 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.9		$E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}% \left(c-1,c-k^{2},c\right),$
ⓘ 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 and $k$ : real or complex modulus Referenced by: §19.25(i), §19.25(i), §19.25(iii), §19.36(i) Permalink: http://dlmf.nist.gov/19.25.E9 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

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		$\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,% c-k^{2},c\right).$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E12 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.13		$D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right).$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $D\left(\NVar{\phi},\NVar{k}\right)$ : incomplete elliptic integral of Legendre’s type, $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.25(i), §19.36(i) Permalink: http://dlmf.nist.gov/19.25.E13 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.14		$\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right)=\tfrac{1}{3}\alpha^{2}R% _{J}\left(c-1,c-k^{2},c,c-\alpha^{2}\right),$
ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.25(i), §19.25(i), §19.25(i), §19.25(iii), §19.6(iv), §19.7(iii), §19.8(i) Permalink: http://dlmf.nist.gov/19.25.E14 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

19.25.15		$\Pi\left(\phi,\alpha^{2},k\right)=R_{-\frac{1}{2}}\left(\tfrac{1}{2},\tfrac{1}% {2},-\tfrac{1}{2},1;c-1,c-k^{2},c,c-\alpha^{2}\right).$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E15 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

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

19.25.16		$\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2% },c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2% })}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})% \right),$
$\omega^{2}=k^{2}/\alpha^{2}$ .
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.25(i) Permalink: http://dlmf.nist.gov/19.25.E16 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

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		$F\left(\phi,k\right)=R_{F}\left(x,y,z\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.15 Permalink: http://dlmf.nist.gov/19.25.E17 Encodings: TeX, pMML, png See also: Annotations for §19.25(i), §19.25 and Ch.19

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

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Keywords:: Bulirsch’s elliptic integrals, Bulirsch’s elliptic integrals, relation to symmetric elliptic integrals, relations to other functions, symmetric elliptic integrals
Notes:: Rewrite Bulirsch’s integrals (§19.2(iii)) in terms of Legendre’s integrals, then use §19.25(i) to convert them to $R$ -functions.
Permalink:: http://dlmf.nist.gov/19.25.ii
See also:: Annotations for §19.25 and Ch.19

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

19.25.19	$\displaystyle\operatorname{cel}\left(k_{c},p,a,b\right)$	$\displaystyle=aR_{F}\left(0,k_{c}^{2},1\right)+\tfrac{1}{3}{(b-pa)}R_{J}\left(% 0,k_{c}^{2},1,p\right),$
ⓘ Symbols: $\operatorname{cel}\left(\NVar{k_{c}},\NVar{p},\NVar{a},\NVar{b}\right)$ : Bulirsch’s complete elliptic integral, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind and $k$ : real or complex modulus Permalink: http://dlmf.nist.gov/19.25.E19 Encodings: TeX, pMML, png See also: Annotations for §19.25(ii), §19.25 and Ch.19
19.25.20	$\displaystyle\operatorname{el1}\left(x,k_{c}\right)$	$\displaystyle=R_{F}\left(r,r+k_{c}^{2},r+1\right),$
ⓘ Symbols: $\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $k$ : real or complex modulus and $r$ : inverse Permalink: http://dlmf.nist.gov/19.25.E20 Encodings: TeX, pMML, png See also: Annotations for §19.25(ii), §19.25 and Ch.19
19.25.21	$\displaystyle\operatorname{el2}\left(x,k_{c},a,b\right)$	$\displaystyle=a\operatorname{el1}\left(x,k_{c}\right)+\tfrac{1}{3}{(b-a)}R_{D}% \left(r,r+k_{c}^{2},r+1\right),$
ⓘ Symbols: $\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\operatorname{el2}\left(\NVar{x},\NVar{k_{c}},\NVar{a},\NVar{b}\right)$ : Bulirsch’s incomplete elliptic integral of the second kind, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $k$ : real or complex modulus and $r$ : inverse Permalink: http://dlmf.nist.gov/19.25.E21 Encodings: TeX, pMML, png See also: Annotations for §19.25(ii), §19.25 and Ch.19
19.25.22	$\displaystyle\operatorname{el3}\left(x,k_{c},p\right)$	$\displaystyle=\operatorname{el1}\left(x,k_{c}\right)+\tfrac{1}{3}{(1-p)}R_{J}% \left(r,r+k_{c}^{2},r+1,r+p\right).$
ⓘ Symbols: $\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\operatorname{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}\right)$ : Bulirsch’s incomplete elliptic integral of the third kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $k$ : real or complex modulus and $r$ : inverse Permalink: http://dlmf.nist.gov/19.25.E22 Encodings: TeX, pMML, png See also: Annotations for §19.25(ii), §19.25 and Ch.19

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

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Keywords:: Legendre’s elliptic integrals, Legendre’s elliptic integrals, relations to other functions, symmetric elliptic integrals, symmetric elliptic integrals
Notes:: Define $c={\csc}^{2}\phi$ , write $\ifrac{(x,y,z,p)}{(z-x)}=(c-1,c-k^{2},c,c-\alpha^{2})$ , then use (19.25.5), (19.25.9), (19.25.14), and (19.25.7) to prove (19.25.24), (19.25.25), (19.25.26), and (19.25.27), respectively.
Referenced by:: Erratum (V1.1.3) for Subsection 19.25(iii)
Permalink:: http://dlmf.nist.gov/19.25.iii
Correction (effective with 1.1.3):: The constraint $(x,y)\neq(0,0)$ was added to the first sentence of this section.
See also:: Annotations for §19.25 and Ch.19

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

19.25.23	$\displaystyle\phi$	$\displaystyle=\operatorname{arccos}\sqrt{\ifrac{x}{z}}=\operatorname{arcsin}% \sqrt{\ifrac{(z-x)}{z}},$
	$\displaystyle k$	$\displaystyle=\sqrt{\frac{z-y}{z-x}},$
	$\displaystyle\alpha^{2}$	$\displaystyle=\frac{z-p}{z-x},$
ⓘ Symbols: $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Permalink: http://dlmf.nist.gov/19.25.E23 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.25(iii), §19.25 and Ch.19

with $\alpha\neq 0$ . Then

19.25.24		$(z-x)^{1/2}R_{F}\left(x,y,z\right)=F\left(\phi,k\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.14(i), §19.25(iii) Permalink: http://dlmf.nist.gov/19.25.E24 Encodings: TeX, pMML, png See also: Annotations for §19.25(iii), §19.25 and Ch.19

19.25.25		$(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F\left(\phi,k\right)-E\left(\phi,% k\right)),$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first 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(iii) Permalink: http://dlmf.nist.gov/19.25.E25 Encodings: TeX, pMML, png See also: Annotations for §19.25(iii), §19.25 and Ch.19

19.25.26		$(z-x)^{3/2}R_{J}\left(x,y,z,p\right)=(3/\alpha^{2}){(\Pi\left(\phi,\alpha^{2},% k\right)-F\left(\phi,k\right))},$
ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.11(i), §19.25(iii) Permalink: http://dlmf.nist.gov/19.25.E26 Encodings: TeX, pMML, png See also: Annotations for §19.25(iii), §19.25 and Ch.19

19.25.27		$2(z-x)^{-1/2}R_{G}\left(x,y,z\right)=E\left(\phi,k\right)+(\cot\phi)^{2}F\left% (\phi,k\right)+(\cot\phi)\sqrt{1-k^{2}{\sin}^{2}\phi}.$
ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\cot\NVar{z}$ : cotangent function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.25(iii) Permalink: http://dlmf.nist.gov/19.25.E27 Encodings: TeX, pMML, png See also: Annotations for §19.25(iii), §19.25 and Ch.19

§19.25(iv) Theta Functions

ⓘ

Keywords:: relations to other functions, symmetric elliptic integrals, theta functions
Permalink:: http://dlmf.nist.gov/19.25.iv
See also:: Annotations for §19.25 and Ch.19

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

§19.25(v) Jacobian Elliptic Functions

ⓘ

Keywords:: Jacobian elliptic functions, interrelations, relations to other functions, symmetric elliptic integrals
Notes:: To prove (19.25.29) use $(\operatorname{cs},\operatorname{ds},\operatorname{ns})=(\operatorname{cn},% \operatorname{dn},1)/\operatorname{sn}$ (suppressing variables (u,k)). For (19.25.30) see Carlson (2006a, Comments following proof of Proposition 4.1). For (19.25.31) see Carlson (2004, (1.8)). In (19.25.32), (19.25.33), and (19.25.34), substitute $x=\operatorname{ps}\left(u,k\right)$ , $\operatorname{sp}\left(u,k\right)$ , and $\operatorname{pq}\left(u,k\right)$ , respectively, to recover (19.25.31).
Referenced by:: §22.15(ii)
Permalink:: http://dlmf.nist.gov/19.25.v
See also:: Annotations for §19.25 and Ch.19

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		$\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $k$ : real or complex modulus Referenced by: §19.25(v) Permalink: http://dlmf.nist.gov/19.25.E30 Encodings: TeX, pMML, png See also: Annotations for §19.25(v), §19.25 and Ch.19

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 (2015, §3.153,1–10 and §3.156,1–9).

§19.25(vi) Weierstrass Elliptic Functions

ⓘ

Keywords:: Weierstrass elliptic functions, Weierstrass elliptic functions, relations to other functions, symmetric elliptic integrals, symmetric elliptic integrals
Notes:: To prove (19.25.35) 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 $\pm$ in (19.25.35) is a consequence of the multivaluedness of the square-roots in (19.16.4). For (19.25.37) we combine its derivative with (19.18.7) and obtain from (19.25.35) that (19.25.37) holds modulo a constant of integration. Combining §23.2(ii) with (19.23.6_5) shows us that both sides of (19.25.37) are $\left(z+2\omega\right)^{-1}+O\left(z+2\omega\right)$ , as $z\to-2\omega$ , and hence the constant of integration is zero. For (19.25.38) and (19.25.39) take $z=-\omega$ in (19.25.35) and (19.25.37), respectively. For (19.25.40) combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)), (19.25.35) and (19.16.1).
Referenced by:: §23.6(iv), Erratum (V1.0.18) for Subsection 19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.vi
Correction (effective with 1.1.0):: This subsection was significantly updated. Equation (19.25.35) has been corrected such that the left-hand side which was originally $z$ , has been replaced by $z+2\omega$ and the right-hand side was multiplied by $\pm 1$ , for some $2\omega\in\mathbb{L}$ . It was then noted that the sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which $\wp\left(z\right)-e_{j}\in(-\infty,0)$ , for some $j$ . Equation (19.25.37) has been corrected such that the left-hand side which was originally $\zeta\left(z\right)+z\wp\left(z\right)$ , has been replaced by $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)$ , the right-hand side has been multiplied by $\pm 1$ . It was then noted that the sign and $\omega$ in (19.25.37) will be the same as in (19.25.35). Equation (19.25.39) has been corrected such that on the left-hand side $\eta_{j}$ was replaced by $\zeta\left(\omega_{j}\right)$ , for some $\omega_{j}$ such that $2\omega_{j}\in\mathbb{L}$ and $\wp\left(\omega_{j}\right)=e_{j}$ . Equation (19.25.40) was corrected such that the left-hand side which was originally $z$ , has been replaced by $z+2\omega$ , and the right-hand side was multiplied by $\pm 1$ , for some $2\omega\in\mathbb{L}$ . It was also mentioned that 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 $j$ .
Correction (effective with 1.0.18):: The links to the Weierstrass lattice roots $e_{j}$ , was incorrectly labeled as the base of the natural logarithm. In order to resolve this inconsistency, we now link the lattice roots $e_{j}$ , and lattice invariants $g_{2}$ , $g_{3}$ , to their respective definitions (see §§23.2(i), 23.3(i)).
Suggested 2018-03-13 by Felix Ospald
See also:: Annotations for §19.25 and Ch.19

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		$\wp\left(z\right)-e_{j}\in\mathbb{C}\setminus(-\infty,0],$
$j=1,2,3.$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{C}$ : complex plane, $\in$ : element of, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction, $j$ : $(=1,2,3)$ and $z$ : complex number Referenced by: (19.25.39) Permalink: http://dlmf.nist.gov/19.25.E36 Encodings: TeX, pMML, png See also: Annotations for §19.25(vi), §19.25 and Ch.19

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

ⓘ

Keywords:: Appell functions, Appell functions, Lauricella’s function, Lauricella’s function, hypergeometric function, hypergeometric function, relation to symmetric elliptic integrals, relations to other functions, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.25.vii
See also:: Annotations for §19.25 and Ch.19

19.25.42		${{}_{2}F_{1}}\left(a,b;c;z\right)=R_{-a}\left(b,c-b;1-z,1\right),$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function Referenced by: §19.20(i), §19.20(iv) Permalink: http://dlmf.nist.gov/19.25.E42 Encodings: TeX, pMML, png See also: Annotations for §19.25(vii), §19.25 and Ch.19

19.25.43		$R_{-a}\left(b_{1},b_{2};z_{1},z_{2}\right)=z_{2}^{-a}{{}_{2}F_{1}}\left(a,b_{1% };b_{1}+b_{2};1-(z_{1}/z_{2})\right).$
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function Referenced by: §19.19 Permalink: http://dlmf.nist.gov/19.25.E43 Encodings: TeX, pMML, png See also: Annotations for §19.25(vii), §19.25 and Ch.19

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.)

19.24 Inequalities 19.26 Addition Theorems

NIST

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