# sin %3F %3C%3D %3F

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## 10 matching pages

##### 1: 19.23 Integral Representations
19.23.1 $R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos^{2}}\theta+z{\sin^{2}}\theta)^% {-1/2}\mathrm{d}\theta,$
19.23.3 $R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos^{2}}\theta+z{\sin^{2}}\theta)% ^{-3/2}{\sin^{2}}\theta\mathrm{d}\theta.$
19.23.5 $R_{F}\left(x,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(x,y{\cos^{2}}% \theta+z{\sin^{2}}\theta\right)\mathrm{d}\theta,$ $\Re y>0$, $\Re z>0$,
In (19.23.8) $n=2$, and in (19.23.9) $n=3$. … With $l_{1},l_{2},l_{3}$ denoting any permutation of $\sin\theta\cos\phi$, $\sin\theta\sin\phi$, $\cos\theta$, …
##### 2: 19.28 Integrals of Elliptic Integrals
19.28.3 $\int_{0}^{1}t^{\sigma-1}(1-t)R_{D}\left(0,t,1\right)\mathrm{d}t=\frac{3}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2}.$
19.28.5 $\int_{z}^{\infty}R_{D}\left(x,y,t\right)\mathrm{d}t=6\!R_{F}\left(x,y,z\right),$
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 $\int_{0}^{\infty}R_{J}\left(tx,y,z,tp\right)\mathrm{d}t=\frac{6}{\sqrt{p}}R_{C% }\left(p,x\right)R_{F}\left(0,y,z\right).$
19.28.9 $\int_{0}^{\pi/2}R_{F}\left({\sin^{2}}\theta{\cos^{2}}\left(x+y\right),{\sin^{2% }}\theta{\cos^{2}}\left(x-y\right),1\right)\mathrm{d}\theta=R_{F}\left(0,{\cos% ^{2}}x,1\right)R_{F}\left(0,{\cos^{2}}y,1\right),$
##### 3: 19.16 Definitions
It should be noted that the integrals (19.16.1)–(19.16.3) have been normalized so that $R_{F}\left(1,1,1\right)=R_{J}\left(1,1,1,1\right)=R_{G}\left(1,1,1\right)=1$. …
19.16.5 $R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\mathrm{d}t}{s(t)(t+z)},$
Just as the elementary function $R_{C}\left(x,y\right)$19.2(iv)) is the degenerate case …and $R_{D}$ is a degenerate case of $R_{J}$, so is $R_{J}$ a degenerate case of the hyperelliptic integral, … (Note that $R_{C}\left(x,y\right)$ is not an elliptic integral.) …
##### 4: 19.2 Definitions
19.2.6 $D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin^{2}}\theta\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{% \sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))% /k^{2}.$
The cases with $\phi=\pi/2$ are the complete integrals: …
###### §19.2(iv) A Related Function: $R_{C}\left(x,y\right)$
When $x$ and $y$ are positive, $R_{C}\left(x,y\right)$ is an inverse circular function if $x and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_{C}\left(x,x\right)$ and $R_{C}\left(0,y\right)$ see (19.6.15). …
##### 5: 19.25 Relations to Other Functions
Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of $R_{D}$; see (19.21.7). … 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 $k\sin\phi$, $-i\tan\phi$, $-ik^{\prime}\tan\phi$, $(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin^{2}}\phi}$, $-ik\sin\phi/\sqrt{1-k^{2}{\sin^{2}}\phi}$. … Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_{F}\left(x,y,z\right)$. … For these results and extensions to the Appell function ${F_{1}}$16.13) and Lauricella’s function $F_{D}$ see Carlson (1963). (${F_{1}}$ and $F_{D}$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.) …
##### 6: 31.2 Differential Equations
###### $F$-Homotopic Transformations
Lastly, $w(z)=(z-a)^{1-\epsilon}w_{3}(z)$ satisfies (31.2.1) if $w_{3}$ is a solution of (31.2.1) with transformed parameters $q_{3}=q+\gamma(1-\epsilon)$; $\alpha_{3}=\alpha+1-\epsilon$, $\beta_{3}=\beta+1-\epsilon$, $\epsilon_{3}=2-\epsilon$. By composing these three steps, there result $2^{3}=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … There are $4!=24$ homographies $\tilde{z}(z)=(Az+B)/(Cz+D)$ that take $0,1,a,\infty$ to some permutation of $0,1,a^{\prime},\infty$, where $a^{\prime}$ may differ from $a$. … There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$-homotopic and homographic transformations. …
##### 7: 19.33 Triaxial Ellipsoids
The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by …
19.33.2 $\frac{S}{2\pi}=c^{2}+\frac{ab}{\sin\phi}\left(E\left(\phi,k\right){\sin^{2}}% \phi+F\left(\phi,k\right){\cos^{2}}\phi\right),$ $a\geq b\geq c$,
and the electric capacity $C=Q/V(0)$ is given by
19.33.6 $1/C=R_{F}\left(a^{2},b^{2},c^{2}\right).$
Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. …
##### 8: 1.6 Vectors and Vector-Valued Functions
The divergence of a differentiable vector-valued function $\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}$ is … The line integral of a vector-valued function $\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}$ along $\mathbf{c}$ is given by … If $C$ is oriented in the positive (anticlockwise) sense, then …Sufficient conditions for this result to hold are that $F_{1}(x,y)$ and $F_{2}(x,y)$ are continuously differentiable on $S$, and $C$ is piecewise differentiable. … For a vector-valued function $\mathbf{F}$, …
##### 9: 31.10 Integral Equations and Representations
for a suitable contour $C$. …The contour $C$ must be such that … for suitable contours $C_{1}$, $C_{2}$. …where $\mathcal{D}_{z}$ is given by (31.10.4). The contours $C_{1}$, $C_{2}$ must be chosen so that …
##### 10: Bibliography S
• I. J. Schoenberg (1971) Norm inequalities for a certain class of ${C}^{\infty}$ functions. Israel J. Math. 10, pp. 364–372.
• R. S. Scorer (1950) Numerical evaluation of integrals of the form $I=\int^{x_{2}}_{x_{1}}f(x)e^{i\phi(x)}dx$ and the tabulation of the function ${\rm Gi}(z)=(1/\pi)\int^{\infty}_{0}{\rm sin}(uz+\frac{1}{3}u^{3})du$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.
• J. Shapiro (1970) Arbitrary $3n-j$ symbols for $\rm{SU}(2)$ . Comput. Phys. Comm. 1 (3), pp. 207–215.
• L. Shen (1998) On an identity of Ramanujan based on the hypergeometric series ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)$ . J. Number Theory 69 (2), pp. 125–134.
• A. Sidi (1997) Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.