# 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 $p$ of $R_{J}$ 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 $x,y,z$ of $R_{F}\left(x,y,z\right)$, $R_{G}\left(x,y,z\right)$, 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 $\Pi\left(\alpha^{2},k\right)-K\left(k\right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(% 0,{k^{\prime}}^{2},1,1-\alpha^{2}\right).$
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 $\wp\left(z\right)-e_{j}<0$, for some $j$. … The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which $\sigma_{j}^{2}(z)<0$, for some $j$. …
##### 4: 19.28 Integrals of Elliptic Integrals
19.28.6 $\int_{0}^{1}R_{D}\left(x,y,v^{2}z+(1-v^{2})p\right)\,\mathrm{d}v=R_{J}\left(x,% y,z,p\right).$
19.28.7 $\int_{0}^{\infty}R_{J}\left(x,y,z,r^{2}\right)\,\mathrm{d}r=\tfrac{3}{2}\pi R_% {F}\left(xy,xz,yz\right),$
19.28.8 $\int_{0}^{\infty}R_{J}\left(tx,y,z,tp\right)\,\mathrm{d}t=\frac{6}{\sqrt{p}}R_% {C}\left(p,x\right)R_{F}\left(0,y,z\right).$