change of parameter of RJ

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1: 19.21 Connection Formulas

… ►

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

… ►Change-of-parameter relations can be used to shift the parameter

p

R_{J}

from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). …

2: 19.15 Advantages of Symmetry

… ►Symmetry in

x, y, z

R_{F}\left(x,y,z\right)

R_{G}\left(x,y,z\right)

, and

R_{J}\left(x,y,z,p\right)

replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). …(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. … ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

3: 19.25 Relations to Other Functions

… ►

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:: pMML, png, TeX
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}

. … ►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). … ►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which