# symmetric elliptic integrals

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## 21—30 of 51 matching pages

##### 21: 19.23 Integral Representations
###### §19.23 Integral Representations
19.23.1 $R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)^% {-1/2}\mathrm{d}\theta,$
19.23.2 $R_{G}\left(0,y,z\right)=\frac{1}{2}\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^% {2}\theta)^{1/2}\mathrm{d}\theta,$
19.23.3 $R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)% ^{-3/2}{\sin}^{2}\theta\mathrm{d}\theta.$
19.23.4 $R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,z{\cos}^{2}% \theta\right)\mathrm{d}\theta=\frac{2}{\pi}\int_{0}^{\infty}R_{C}\left(y{\cosh% }^{2}t,z\right)\mathrm{d}t.$
##### 22: 19.29 Reduction of General Elliptic Integrals
###### §19.29(i) Reduction Theorems
19.29.33 $(x-y)^{2}U=\left(\sqrt{x^{4}+a^{4}}+\sqrt{y^{4}+a^{4}}\right)^{2}-(x^{2}-y^{2}% )^{2}.$
##### 24: 19.19 Taylor and Related Series
###### §19.19 Taylor and Related Series
19.19.6 $R_{J}\left(x,y,z,p\right)=R_{-\frac{3}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},% \tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z,p,p\right)$
##### 25: 19.30 Lengths of Plane Curves
19.30.3 $s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},$
19.30.9 $s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},$ $r=b^{4}/y^{2}$.
19.30.11 $s=2a^{2}\int_{0}^{r}\frac{\mathrm{d}t}{\sqrt{4a^{4}-t^{4}}}=\sqrt{2a^{2}}R_{F}% \left(q-1,q,q+1\right),$ $q=2a^{2}/r^{2}=\sec\left(2\theta\right)$,
19.30.13 $P=4\sqrt{2a^{2}}R_{F}\left(0,1,2\right)=\sqrt{2a^{2}}\times 5.24411\;51\ldots=% 4aK\left(1/\sqrt{2}\right)=a\times 7.41629\;87\dots.$
##### 26: 20.9 Relations to Other Functions
20.9.3 $R_{F}\left(\frac{{\theta_{2}}^{2}\left(z,q\right)}{{\theta_{2}}^{2}\left(0,q% \right)},\frac{{\theta_{3}}^{2}\left(z,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},\frac{{\theta_{4}}^{2}\left(z,q\right)}{{\theta_{4}}^{2}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,$
##### 27: 19.36 Methods of Computation
###### §19.36 Methods of Computation
19.36.3 $R_{F}\left(1,2,4\right)=R_{F}\left(z_{1},z_{2},z_{3}\right),$
19.36.5 $R_{F}\left(1,2,4\right)=0.68508\;58166\dots.$
19.36.8 $R_{F}\left(t_{n}^{2},t_{n}^{2}+\theta c_{n}^{2},t_{n}^{2}+\theta a_{n}^{2}\right)$
19.36.11 $R_{F}\left(1,2,4\right)=R_{C}\left(T^{2}+M^{2},T^{2}\right)=0.68508\;58166,$
##### 28: 22.15 Inverse Functions
For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …
##### 29: Bibliography Z
• D. G. Zill and B. C. Carlson (1970) Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.
• ##### 30: Bibliography N
• E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.