# symmetric elliptic integrals

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##### 1: 19.16 Definitions
###### §19.16(i) SymmetricIntegrals
19.16.5 $R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\,\mathrm{d}t}{s(t)(t+z)},$
which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\dots,n$. … …
##### 2: 19.15 Advantages of Symmetry
For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …
##### 5: 19.21 Connection Formulas
###### §19.21 Connection Formulas
19.21.3 $6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))=3% zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right).$
19.21.11 $6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),$
###### §19.21(iii) Change of Parameter of $R_{J}$
19.21.15 $pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),$ $pq=yz$.
19.22.2 $2R_{G}\left(0,x^{2},y^{2}\right)=4R_{G}\left(0,xy,a^{2}\right)-xyR_{F}\left(0,% xy,a^{2}\right),$
##### 7: 19.18 Derivatives and Differential Equations
###### §19.18(i) Derivatives
19.18.2 $\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}% \left(x+a,x+b,x+c\right).$
###### §19.18(ii) Differential Equations
and two similar equations obtained by permuting $x,y,z$ in (19.18.10). … The next four differential equations apply to the complete case of $R_{F}$ and $R_{G}$ in the form $R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)$ (see (19.16.20) and (19.16.23)). …
##### 10: 19.24 Inequalities
###### §19.24(i) Complete Integrals
19.24.7 $L(a,b)=8R_{G}\left(0,a^{2},b^{2}\right).$