19 Elliptic Integrals Symmetric Integrals 19.20 Special Cases 19.22 Quadratic Transformations

§19.21 Connection Formulas

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Keywords:: connection formulas, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.21
See also:: Annotations for Ch.19

§19.21(i) Complete Integrals
§19.21(ii) Incomplete Integrals
§19.21(iii) Change of Parameter of $R_{J}$

§19.21(i) Complete Integrals

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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)).
Permalink:: http://dlmf.nist.gov/19.21.i
See also:: Annotations for §19.21 and Ch.19

Legendre’s relation (19.7.1) can be written

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

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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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $(\NVar{a},\NVar{b}]$ : 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
Encodings:: TeX, pMML, png
See also:: Annotations for §19.21(i), §19.21 and Ch.19

The case shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is $\pi/4$ .

19.21.2 $3R_{F}\left(0,y,z\right)=zR_{D}\left(0,y,z\right)+yR_{D}\left(0,z,y\right).$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(i)
Permalink:: http://dlmf.nist.gov/19.21.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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

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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 and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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

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

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/19.21.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §19.21(i), §19.21 and Ch.19

19.21.5 $2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)\pm\mathrm{i}2R_{G}\left(% 0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)\mp\mathrm{i}zR_{F}\left(0,z,1% \right).$

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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 and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/19.21.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §19.21(i), §19.21 and Ch.19

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

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

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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 and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : 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
See also:: Annotations for §19.21(i), §19.21 and Ch.19

If and , then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1).

§19.21(ii) Incomplete Integrals

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Notes:: See Carlson (1977b, (5.9-5), (5.9-6)) and (19.20.25). To obtain (19.21.7) eliminate $R_{D}\left(z,x,y\right)$ 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 (19.16.2_5) and similarly for and . Then use (19.21.9).
Permalink:: http://dlmf.nist.gov/19.21.ii
See also:: Annotations for §19.21 and Ch.19

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

19.21.7 $(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z% \right)-3y^{1/2}x^{-1/2}z^{-1/2},$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.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 §19.21(ii), §19.21 and Ch.19

or the corresponding equation with and interchanged.

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

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.E8
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 §19.21(ii), §19.21 and Ch.19

19.21.9 $xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,y,z\right)=3R_% {F}\left(x,y,z\right).$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

19.21.10 $2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}% \left(x,y,z\right)+x^{1/2}y^{1/2}z^{-1/2},$ $z\neq 0$ .

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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 and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)
Permalink:: http://dlmf.nist.gov/19.21.E10
Encodings:: TeX, pMML, png
Correction (effective with 1.1.7):: An incorrect factor of 3 was removed in the last term on the right-hand side.
Suggested 2022-06-27 by Abdulhafeez Abdulsalam
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 §19.21(ii), §19.21 and Ch.19

Because $R_{G}$ 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 $R_{G}$ is calculated from $R_{F}$ and $R_{D}$ (see §19.36(i)).

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

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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 and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

where both summations extend over the three cyclic permutations of .

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

§19.21(iii) Change of Parameter of $R_{J}$

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Keywords:: change of parameter of $R_{J}$ , circular cases, hyperbolic cases, symmetric elliptic integrals
Notes:: See Zill and Carlson (1970, (4.6)).
Referenced by:: §19.7(iii)
Permalink:: http://dlmf.nist.gov/19.21.iii
See also:: Annotations for §19.21 and Ch.19

Let be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter of $R_{J}$ 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 $(p-x)R_{J}\left(x,y,z,p\right)+(q-x)R_{J}\left(x,y,z,q\right)=3R_{F}\left(x,y,% z\right)-3R_{C}\left(\xi,\eta\right),$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\xi$ 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
Encodings:: TeX, pMML, png
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

where

19.21.13

$\xi=yz/x,$

$\eta=pq/x,$

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Symbols:: $\xi$ and $\eta$
Permalink:: http://dlmf.nist.gov/19.21.E13
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

and may be permuted. Also,

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

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Symbols:: $\xi$ and $\eta$
Permalink:: http://dlmf.nist.gov/19.21.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

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 $R_{C}\left(\xi,\eta\right)=0$ ; hence