§19.25 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/19.25

§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

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 $\mathop{R_{G}\/}\nolimits$ 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).
Keywords:: Legendre’s elliptic integrals, symmetric elliptic integrals
Referenced by:: §19.25(ii), ¶ ‣ §19.36(i), §19.7(ii)
Permalink:: http://dlmf.nist.gov/19.25.i

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

19.25.1

$\mathop{K\/}\nolimits\!\left(k\right)=\mathop{R_{F}\/}\nolimits\!\left(0,{k^{{\prime}}}^{2},1\right),$

$\mathop{E\/}\nolimits\!\left(k\right)=2\!\mathop{R_{G}\/}\nolimits\!\left(0,{k^{{\prime}}}^{2},1\right),$

$\mathop{E\/}\nolimits\!\left(k\right)=\tfrac{1}{3}{k^{{\prime}}}^{2}\left(\mathop{R_{D}\/}\nolimits\!\left(0,{k^{{\prime}}}^{2},1\right)+\mathop{R_{D}\/}\nolimits\!\left(0,1,{k^{{\prime}}}^{2}\right)\right),$

$\mathop{K\/}\nolimits\!\left(k\right)-\mathop{E\/}\nolimits\!\left(k\right)=k^{2}\mathop{D\/}\nolimits\!\left(k\right)=\tfrac{1}{3}k^{2}\mathop{R_{D}\/}\nolimits\!\left(0,{k^{{\prime}}}^{2},1\right),$

$\mathop{E\/}\nolimits\!\left(k\right)-{k^{{\prime}}}^{2}\mathop{K\/}\nolimits\!\left(k\right)=\tfrac{1}{3}k^{2}{k^{{\prime}}}^{2}\mathop{R_{D}\/}\nolimits\!\left(0,1,{k^{{\prime}}}^{2}\right).$

19.25.2 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)-\mathop{K\/}\nolimits\!\left(k\right)=\tfrac{1}{3}\alpha^{2}\mathop{R_{J}\/}\nolimits\!\left(0,{k^{{\prime}}}^{2},1,1-\alpha^{2}\right).$

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : 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
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19.25.3 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\tfrac{1}{2}\pi\mathop{R_{{-\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},-\tfrac{1}{2},1;{k^{{\prime}}}^{2},1,1-\alpha^{2}\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : 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
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with Cauchy principal value

19.25.4 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/\alpha^{2})\mathop{R_{J}\/}\nolimits\!\left(0,1-k^{2},1,1-(k^{2}/\alpha^{2})\right),$ $-\infty<k^{2}<1<\alpha^{2}$ .

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : 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
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19.25.5 $\mathop{F\/}\nolimits\!\left(\phi,k\right)=\mathop{R_{F}\/}\nolimits\!\left(c-1,c-k^{2},c\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : 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, ¶ ‣ §19.36(ii), §19.6(ii), §19.9(ii), §19.9(ii)
Permalink:: http://dlmf.nist.gov/19.25.E5
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19.25.6 $\frac{\partial\mathop{F\/}\nolimits\!\left(\phi,k\right)}{\partial k}=\tfrac{1}{3}k\mathop{R_{D}\/}\nolimits\!\left(c-1,c,c-k^{2}\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i)
Permalink:: http://dlmf.nist.gov/19.25.E6
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19.25.7 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=2\!\mathop{R_{G}\/}\nolimits\!\left(c-1,c-k^{2},c\right)-(c-1)\mathop{R_{F}\/}\nolimits\!\left(c-1,c-k^{2},c\right)-\sqrt{(c-1)(c-k^{2})/c},$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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, ¶ ‣ §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.25.E7
Encodings:: TeX, pMML, png

19.25.8 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=\mathop{R_{{-\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{3}{2};c-1,c-k^{2},c\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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

19.25.9 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=\mathop{R_{F}\/}\nolimits\!\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}\mathop{R_{D}\/}\nolimits\!\left(c-1,c-k^{2},c\right),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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
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19.25.10 $\mathop{E\/}\nolimits\!\left(\phi,k\right)={k^{{\prime}}}^{2}\mathop{R_{F}\/}\nolimits\!\left(c-1,c-k^{2},c\right)+\tfrac{1}{3}k^{2}{k^{{\prime}}}^{2}\mathop{R_{D}\/}\nolimits\!\left(c-1,c,c-k^{2}\right)+k^{2}\sqrt{(c-1)/(c(c-k^{2}))},$ $c>k^{2}$ ,

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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)
Permalink:: http://dlmf.nist.gov/19.25.E10
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19.25.11 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=-\tfrac{1}{3}{k^{{\prime}}}^{2}\mathop{R_{D}\/}\nolimits\!\left(c-k^{2},c,c-1\right)+\sqrt{(c-k^{2})/(c(c-1))},$ $\phi\neq\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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)
Permalink:: http://dlmf.nist.gov/19.25.E11
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Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of $\mathop{R_{D}\/}\nolimits$ ; 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 $\frac{\partial\mathop{E\/}\nolimits\!\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}k\mathop{R_{D}\/}\nolimits\!\left(c-1,c-k^{2},c\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\phi$ : 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

19.25.13 $\mathop{D\/}\nolimits\!\left(\phi,k\right)=\tfrac{1}{3}\mathop{R_{D}\/}\nolimits\!\left(c-1,c-k^{2},c\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{D\/}\nolimits\!\left(\phi,k\right)$ : 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
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19.25.14 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)-\mathop{F\/}\nolimits\!\left(\phi,k\right)=\tfrac{1}{3}\alpha^{2}\mathop{R_{J}\/}\nolimits\!\left(c-1,c-k^{2},c,c-\alpha^{2}\right),$

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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

19.25.15 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)=\mathop{R_{{-\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},1;c-1,c-k^{2},c,c-\alpha^{2}\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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

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

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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
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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 $\mathop{F\/}\nolimits\!\left(\phi,k\right)=\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : 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

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

The three changes of parameter of $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ 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

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.
Keywords:: Bulirsch’s elliptic integrals, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.25.ii

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

19.25.19 $\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,a,b\right)=a\mathop{R_{F}\/}\nolimits\!\left(0,k_{c}^{2},1\right)+\tfrac{1}{3}{(b-pa)}\mathop{R_{J}\/}\nolimits\!\left(0,k_{c}^{2},1,p\right),$

Symbols:: $\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,a,b\right)$ : Bulirsch’s complete elliptic integral, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.25.E19
Encodings:: TeX, pMML, png

19.25.20 $\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)=\mathop{R_{F}\/}\nolimits\!\left(r,r+k_{c}^{2},r+1\right),$

Symbols:: $\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $k$ : real or complex modulus and $r$ : inverse
Permalink:: http://dlmf.nist.gov/19.25.E20
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19.25.21 $\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)=a\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)+\tfrac{1}{3}{(b-a)}\mathop{R_{D}\/}\nolimits\!\left(r,r+k_{c}^{2},r+1\right),$

Symbols:: $\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)$ : Bulirsch’s incomplete elliptic integral of the second kind, $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.25.22 $\mathop{\mathrm{el3}\/}\nolimits\!\left(x,k_{c},p\right)=\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)+\tfrac{1}{3}{(1-p)}\mathop{R_{J}\/}\nolimits\!\left(r,r+k_{c}^{2},r+1,r+p\right).$

Symbols:: $\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\mathop{\mathrm{el3}\/}\nolimits\!\left(x,k_{c},p\right)$ : Bulirsch’s incomplete elliptic integral of the third kind, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $k$ : real or complex modulus and $r$ : inverse
Permalink:: http://dlmf.nist.gov/19.25.E22
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§19.25(iii) Symmetric Integrals as Legendre’s Integrals

Notes:: Define $c={\mathop{\csc\/}\nolimits^{{2}}}\phi$ , 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.
Keywords:: Legendre’s elliptic integrals, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.25.iii

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

19.25.23

$\phi=\mathop{\mathrm{arccos}\/}\nolimits\sqrt{\ifrac{x}{z}}=\mathop{\mathrm{arcsin}\/}\nolimits\sqrt{\ifrac{(z-x)}{z}},$

$k=\sqrt{\frac{z-y}{z-x}},$

$\alpha^{2}=\frac{z-p}{z-x},$

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : 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

with $\alpha\neq 0$ . Then

19.25.24 $(z-x)^{{1/2}}\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\mathop{F\/}\nolimits\!\left(\phi,k\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : 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
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19.25.25 $(z-x)^{{3/2}}\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=(3/k^{2})(\mathop{F\/}\nolimits\!\left(\phi,k\right)-\mathop{E\/}\nolimits\!\left(\phi,k\right)),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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
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19.25.26 $(z-x)^{{3/2}}\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=(3/\alpha^{2}){(\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)-\mathop{F\/}\nolimits\!\left(\phi,k\right))},$

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : 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
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19.25.27 $2(z-x)^{{-1/2}}\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\mathop{E\/}\nolimits\!\left(\phi,k\right)+(\mathop{\cot\/}\nolimits\phi)^{2}\mathop{F\/}\nolimits\!\left(\phi,k\right)+(\mathop{\cot\/}\nolimits\phi)\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}.$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : 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
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§19.25(iv) Theta Functions

Keywords:: symmetric elliptic integrals, theta functions
Permalink:: http://dlmf.nist.gov/19.25.iv

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

§19.25(v) Jacobian Elliptic Functions

Notes:: To prove (19.25.29) use $(\mathop{\mathrm{cs}\/}\nolimits,\mathop{\mathrm{ds}\/}\nolimits,\mathop{\mathrm{ns}\/}\nolimits)=(\mathop{\mathrm{cn}\/}\nolimits,\mathop{\mathrm{dn}\/}\nolimits,1)/\mathop{\mathrm{sn}\/}\nolimits$ (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 $x=\mathop{\mathrm{ps}\/}\nolimits\left(u,k\right)$ , $\mathop{\mathrm{sp}\/}\nolimits\left(u,k\right)$ , and $\mathop{\mathrm{pq}\/}\nolimits\left(u,k\right)$ , respectively, to recover (19.25.31).
Keywords:: Jacobian elliptic functions, symmetric elliptic integrals
Referenced by:: §22.15(ii)
Permalink:: http://dlmf.nist.gov/19.25.v

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 $\Delta(\mathrm{p,q})={\mathop{\mathrm{ps}\/}\nolimits^{{2}}}\left(u,k\right)-{\mathop{\mathrm{qs}\/}\nolimits^{{2}}}\left(u,k\right)=-\Delta(\mathrm{q,p}),$

Symbols:: $\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)$ : generic Jacobian elliptic function, $k$ : real or complex modulus and $\Delta(\mathrm{p,q})$ : function
Permalink:: http://dlmf.nist.gov/19.25.E28
Encodings:: TeX, pMML, png

which implies

19.25.29

$\Delta(\mathrm{n,d})=k^{2},$

$\Delta(\mathrm{d,c})={k^{{\prime}}}^{2},$

$\Delta(\mathrm{n,c})=1.$

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

If ${\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(u,k\right)\geq 0$ , then

19.25.30 $\mathop{\mathrm{am}\/}\nolimits\left(u,k\right)=\mathop{R_{C}\/}\nolimits\!\left({\mathop{\mathrm{cs}\/}\nolimits^{{2}}}\left(u,k\right),{\mathop{\mathrm{ns}\/}\nolimits^{{2}}}\left(u,k\right)\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : 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

19.25.31 $u=\mathop{R_{F}\/}\nolimits\!\left({\mathop{\mathrm{ps}\/}\nolimits^{{2}}}\left(u,k\right),{\mathop{\mathrm{qs}\/}\nolimits^{{2}}}\left(u,k\right),{\mathop{\mathrm{rs}\/}\nolimits^{{2}}}\left(u,k\right)\right);$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)$ : 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

compare (19.25.35) and (20.9.3).

19.25.32 $\mathop{\mathrm{arcps}\/}\nolimits\left(x,k\right)=\mathop{R_{F}\/}\nolimits\!\left(x^{2},x^{2}+\Delta(\mathrm{q,p}),x^{2}+\Delta(\mathrm{r,p})\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.25.33 $\mathop{\mathrm{arcsp}\/}\nolimits\left(x,k\right)=x\mathop{R_{F}\/}\nolimits\!\left(1,1+\Delta(\mathrm{q,p})x^{2},1+\Delta(\mathrm{r,p})x^{2}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.25.34 $\mathop{\mathrm{arcpq}\/}\nolimits\left(x,k\right)=\sqrt{w}\mathop{R_{F}\/}\nolimits\!\left(x^{2},1,1+\Delta(\mathrm{r,q})w\right),$ $w=\ifrac{(1-x^{2})}{\Delta(\mathrm{q,p})}$ ,

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

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

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 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ . See (19.29.19), Carlson (2005), and (22.15.11), and compare with Abramowitz and Stegun (1964, Eqs. (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

Notes:: To prove (19.25.35) use (23.6.36), with $z=\mathop{\wp\/}\nolimits\!\left(w\right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\mathop{\wp\/}\nolimits\!\left(w\right)$ and compare with (19.16.1). Then put $z=\omega _{j}$ to obtain (19.25.38). For (19.25.37) and (19.25.39) see Carlson (1964, (3.10) and (3.2)). For (19.25.40) combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)) and (19.25.35).
Keywords:: Weierstrass elliptic functions, symmetric elliptic integrals
Referenced by:: ¶ ‣ §23.6(iv)
Permalink:: http://dlmf.nist.gov/19.25.vi

For the notation see §23.2.

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

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function and $e$ : base of exponential function
Referenced by:: §19.25(v), §19.25(vi), §20.9(i)
Permalink:: http://dlmf.nist.gov/19.25.E35
Encodings:: TeX, pMML, png

provided that

19.25.36 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{j}\in\Complex\setminus(-\infty,0],$ $j=1,2,3$ ,

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $e$ : base of exponential function, $(a,b]$ : half-closed interval and $\setminus$ : set subtraction
Permalink:: http://dlmf.nist.gov/19.25.E36
Encodings:: TeX, pMML, png

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

19.25.37 $\mathop{\zeta\/}\nolimits\!\left(z\right)+z\mathop{\wp\/}\nolimits\!\left(z\right)=2\!\mathop{R_{G}\/}\nolimits\!\left(\mathop{\wp\/}\nolimits\!\left(z\right)-e_{1},\mathop{\wp\/}\nolimits\!\left(z\right)-e_{2},\mathop{\wp\/}\nolimits\!\left(z\right)-e_{3}\right).$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function and $e$ : base of exponential function
Referenced by:: §19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E37
Encodings:: TeX, pMML, png

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

19.25.38 $\omega _{j}=\mathop{R_{F}\/}\nolimits\!\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $e$ : base of exponential function and $k$ : real or complex modulus
Referenced by:: §19.25(vi), §19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E38
Encodings:: TeX, pMML, png

19.25.39 $\eta _{j}+\omega _{j}e_{j}=2\!\mathop{R_{G}\/}\nolimits\!\left(0,e_{j}-e_{k},e_{j}-e_{\ell}\right).$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $e$ : base of exponential function and $k$ : real or complex modulus
Referenced by:: §19.25(vi), §19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E39
Encodings:: TeX, pMML, png

Lastly,

19.25.40 $z=\mathop{\sigma\/}\nolimits\!\left(z\right)\mathop{R_{F}\/}\nolimits\!\left(\sigma _{1}^{2}(z),\sigma _{2}^{2}(z),\sigma _{3}^{2}(z)\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function and $\sigma _{j}(z)$ : function
Referenced by:: §19.25(vi)
Permalink:: http://dlmf.nist.gov/19.25.E40
Encodings:: TeX, pMML, png

where

19.25.41 $\sigma _{j}(z)=\mathop{\exp\/}\nolimits\!\left(-\eta _{j}z\right)\mathop{\sigma\/}\nolimits\!\left(z+\omega _{j}\right)/\mathop{\sigma\/}\nolimits\!\left(\omega _{j}\right),$ $j=1,2,3$ .

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function and $\sigma _{j}(z)$ : function
Permalink:: http://dlmf.nist.gov/19.25.E41
Encodings:: TeX, pMML, png

§19.25(vii) Hypergeometric Function

Keywords:: Appell functions, Lauricella’s function, hypergeometric function, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.25.vii

19.25.42 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b;c;z\right)=\mathop{R_{{-a}}\/}\nolimits\!\left(b,c-b;1-z,1\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function
Referenced by:: §19.20(i), §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.25.E42
Encodings:: TeX, pMML, png

19.25.43 $\mathop{R_{{-a}}\/}\nolimits\!\left(b_{1},b_{2};z_{1},z_{2}\right)=z_{2}^{{-a}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b_{1};b_{1}+b_{2};1-(z_{1}/z_{2})\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.25.E43
Encodings:: TeX, pMML, png

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