19 Elliptic Integrals Symmetric Integrals 19.20 Special Cases 19.22 Quadratic Transformations

§19.21 Connection Formulas

Keywords:: symmetric elliptic integrals
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§19.21(i) Complete Integrals
§19.21(ii) Incomplete Integrals
§19.21(iii) Change of Parameter of $\mathop{R_{J}\/}\nolimits$

§19.21(i) Complete Integrals

Notes:: To prove (19.21.1) see the text following (19.21.6), use (19.20.10), and analytic continuation. For (19.21.2) put in (19.21.9). For (19.21.3) put in (19.21.11) and (19.21.10). (19.21.6) is equivalent to Zill and Carlson (1970, (7.15)).
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Legendre’s relation (19.7.1) can be written

19.21.1 $\mathop{R_{F}\/}\nolimits\!\left(0,z+1,z\right)\mathop{R_{D}\/}\nolimits\!% \left(0,z+1,1\right)+\mathop{R_{D}\/}\nolimits\!\left(0,z+1,z\right)\mathop{R_% {F}\/}\nolimits\!\left(0,z+1,1\right)=3\pi/(2z),$ $z\in\Complex\setminus(-\infty,0]$ .

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, $\Complex$ : complex plane (excluding infinity), $\in$ : element of, : half-closed interval and $\setminus$ : set subtraction
Referenced by:: §19.21(i), §19.21(i), §19.7(i)
Permalink:: http://dlmf.nist.gov/19.21.E1
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The case shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is $\pi/4$ .

19.21.2 $3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)=z\mathop{R_{D}\/}\nolimits\!% \left(0,y,z\right)+y\mathop{R_{D}\/}\nolimits\!\left(0,z,y\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(i)
Permalink:: http://dlmf.nist.gov/19.21.E2
Encodings:: TeX, pMML, png

19.21.3 $6\!\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)=yz(\mathop{R_{D}\/}\nolimits% \!\left(0,y,z\right)+\mathop{R_{D}\/}\nolimits\!\left(0,z,y\right))=3z\mathop{% R_{F}\/}\nolimits\!\left(0,y,z\right)+z(y-z)\mathop{R_{D}\/}\nolimits\!\left(0% ,y,z\right).$

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 and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.22(i), §19.34
Permalink:: http://dlmf.nist.gov/19.21.E3
Encodings:: TeX, pMML, png

The complete cases of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{G}\/}\nolimits$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if $0<\mathop{\mathrm{ph}\/}\nolimits z<\pi$ , and lower signs if $-\pi<\mathop{\mathrm{ph}\/}\nolimits z<0$ :

19.21.4 $\mathop{R_{F}\/}\nolimits\!\left(0,z-1,z\right)=\mathop{R_{F}\/}\nolimits\!% \left(0,1-z,1\right)\mp i\!\mathop{R_{F}\/}\nolimits\!\left(0,z,1\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Permalink:: http://dlmf.nist.gov/19.21.E4
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19.21.5 $2\!\mathop{R_{G}\/}\nolimits\!\left(0,z-1,z\right)=2\!\mathop{R_{G}\/}% \nolimits\!\left(0,1-z,1\right)\pm i2\!\mathop{R_{G}\/}\nolimits\!\left(0,z,1% \right)+(z-1)\mathop{R_{F}\/}\nolimits\!\left(0,1-z,1\right)\mp iz\!\mathop{R_% {F}\/}\nolimits\!\left(0,z,1\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Permalink:: http://dlmf.nist.gov/19.21.E5
Encodings:: TeX, pMML, png

Let , , and be positive and distinct, and permute and to ensure that does not lie between and . The complete case of $\mathop{R_{J}\/}\nolimits$ can be expressed in terms of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{D}\/}\nolimits$ :

19.21.6 $(\sqrt{rp}/z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)={(r-1)}\mathop{R_% {F}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{D}\/}\nolimits\!\left(p,rz,z% \right)+\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{F}\/}\nolimits% \!\left(p,rz,z\right),$

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 and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.21(i), §19.21(i)
Permalink:: http://dlmf.nist.gov/19.21.E6
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If and , then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1).

§19.21(ii) Incomplete Integrals

Notes:: See Carlson (1977b, (5.9-5),(5.9-6)) and (19.20.25). To obtain (19.21.7) eliminate $\mathop{R_{D}\/}\nolimits\!\left(z,x,y\right)$ between (19.21.8) and (19.21.9), which follow from Carlson (1977b, (5.9-5, (6.6-5), and (5.9-6)). For (19.21.10) see Carlson (1977b, Table 9.3-1). To prove (19.21.11) write $xt/(t+x)=x-(x^{2}/(t+x))$ in (19.23.7) and similarly for and . Then use (19.21.9).
Permalink:: http://dlmf.nist.gov/19.21.ii

$\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ is symmetric only in and , but either (nonzero) or (nonzero) can be moved to the third position by using

19.21.7 $(x-y)\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+(z-y)\mathop{R_{D}\/}% \nolimits\!\left(x,y,z\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)% -3\sqrt{y/(xz)},$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(ii), §19.25(i)
Permalink:: http://dlmf.nist.gov/19.21.E7
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or the corresponding equation with and interchanged.

19.21.8 $\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+\mathop{R_{D}\/}\nolimits\!\left% (z,x,y\right)+\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=3(xyz)^{{-1/2}},$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.21(ii), §19.33(iii)
Permalink:: http://dlmf.nist.gov/19.21.E8
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19.21.9 $x\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+y\mathop{R_{D}\/}\nolimits\!% \left(z,x,y\right)+z\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=3\!\mathop{R% _{F}\/}\nolimits\!\left(x,y,z\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(i), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.21.E9
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19.21.10 $2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=z\mathop{R_{F}\/}\nolimits\!% \left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)\mathop{R_{D}\/}\nolimits\!\left(x,y,% z\right)+\sqrt{xy/z},$ $z\neq 0$ .

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 and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), ¶ ‣ §19.36(i), ¶ ‣ §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.21.E10
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Because $\mathop{R_{G}\/}\nolimits$ is completely symmetric, can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\leq 0$ if the variables are real, thereby avoiding cancellations when $\mathop{R_{G}\/}\nolimits$ is calculated from $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{D}\/}\nolimits$ (see §19.36(i)).

19.21.11 $6\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=3(x+y+z)\mathop{R_{F}\/}% \nolimits\!\left(x,y,z\right)-\sum x^{2}\mathop{R_{D}\/}\nolimits\!\left(y,z,x% \right)=\sum x(y+z)\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right),$

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 and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.21.E11
Encodings:: TeX, pMML, png

where both summations extend over the three cyclic permutations of .

Connection formulas for $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ are given in Carlson (1977b, pp. 99, 101, and 123–124).

§19.21(iii) Change of Parameter of $\mathop{R_{J}\/}\nolimits$

Notes:: See Zill and Carlson (1970, (4.6)).
Keywords:: symmetric elliptic integrals
Referenced by:: §19.7(iii)
Permalink:: http://dlmf.nist.gov/19.21.iii

Let be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter of $\mathop{R_{J}\/}\nolimits$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). The latter case allows evaluation of Cauchy principal values (see (19.20.14)).

19.21.12 $(p-x)\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)+(q-x)\mathop{R_{J}\/}% \nolimits\!\left(x,y,z,q\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z% \right)-3\!\mathop{R_{C}\/}\nolimits\!\left(\xi,\eta\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\xi$ and $\eta$
Referenced by:: §19.15, §19.20(iii), §19.21(iii), §19.25(i), §19.26(i)
Permalink:: http://dlmf.nist.gov/19.21.E12
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where

19.21.13

$\xi=yz/x,$

$\eta=pq/x,$

Symbols:: $\xi$ and $\eta$
Permalink:: http://dlmf.nist.gov/19.21.E13
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

and may be permuted. Also,

19.21.14 $\eta-\xi=p+q-y-z=\frac{(p-y)(p-z)}{p-x}=\frac{(q-y)(q-z)}{q-x}=\frac{(p-y)(q-y% )}{x-y}=\frac{(p-z)(q-z)}{x-z}.$

Symbols:: $\xi$ and $\eta$
Permalink:: http://dlmf.nist.gov/19.21.E14
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For each value of , permutation of produces three values of , one of which lies in the same region as and two lie in the other region of the same type. In (19.21.12), if is the largest (smallest) of , and , then and lie in the same region if it is circular (hyperbolic); otherwise and lie in different regions, both circular or both hyperbolic. If , then $\xi=\eta=\infty$ and $\mathop{R_{C}\/}\nolimits\!\left(\xi,\eta\right)=0$ ; hence

19.21.15 $p\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)+q\mathop{R_{J}\/}\nolimits\!% \left(0,y,z,q\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.21.E15
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.20 Special Cases 19.22 Quadratic Transformations

§19.21 Connection Formulas

Contents

§19.21(i) Complete Integrals

§19.21(ii) Incomplete Integrals

§19.21(iii) Change of Parameter of

§19.21(iii) Change of Parameter of $\mathop{R_{J}\/}\nolimits$