§19.21 Connection Formulas

Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.21

§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 $x=0$ in (19.21.9). For (19.21.3) put $x=0$ in (19.21.11) and (19.21.10). (19.21.6) is equivalent to Zill and Carlson (1970, (7.15)).
Permalink:: http://dlmf.nist.gov/19.21.i

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, $(a,b]$ : 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 $z=1$ 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 $y$ , $z$ , and $p$ be positive and distinct, and permute $y$ and $z$ to ensure that $y$ does not lie between $z$ and $p$ . 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 $(\sqrt{rp}/z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)={(r-1)}\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{D}\/}\nolimits\!\left(p,rz,z\right)+\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{F}\/}\nolimits\!\left(p,rz,z\right),$ $r=(y-p)/(y-z)>0$ .

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 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.21(i), §19.21(i)
Permalink:: http://dlmf.nist.gov/19.21.E6
Encodings:: TeX, pMML, png

If $0<p<z$ and $y=z+1$ , 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 $y$ and $z$ . 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 $x$ and $y$ , but either (nonzero) $x$ or (nonzero) $y$ 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
Encodings:: TeX, pMML, png

or the corresponding equation with $x$ and $y$ 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
Encodings:: TeX, pMML, png

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
Encodings:: TeX, pMML, png

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, $x,y,z$ 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 $x,y,z$ .

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 $x,y,z$ be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter $p$ 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

$(p-x)(q-x)=(y-x)(z-x),$

$\xi=yz/x,$

$\eta=pq/x,$

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

and $x,y,z$ 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
Encodings:: TeX, pMML, png

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

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

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

§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 $\mathop{R_{J}\/}\nolimits$