change of parameter of RJ

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1: 19.21 Connection Formulas

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§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 $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ 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

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19.25.2 $$\mathrm{\Pi }({\alpha }^2,k)-K\left(k\right)=\frac{1}{3}{\alpha }^2{R}_J(0,k_{}^{\prime }{}_{}{}^2,1,1-{\alpha }^2).$$

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Symbols:

$R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Pi }({\alpha }^2,k)$: 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:

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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 $\sin \varphi$ ($=\sqrt{(z-x)/z}$) into $k\sin \varphi$, $-\mathrm{i}\tan \varphi$, $-\mathrm{i}{k}^{\prime }\tan \varphi$, $(k^{\prime }\sin \varphi )/\sqrt{1-k^2{\sin }^2\varphi }$, $-\mathrm{i}k\sin \varphi /\sqrt{1-k^2{\sin }^2\varphi }$. … ►The three changes of parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ 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 , for some $j$. … ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which , for some $j$. …

4: 19.28 Integrals of Elliptic Integrals

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19.28.1 $${\int }_0^1{t}^{\sigma -1}{R}_F(0,t,1)dt=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E1

Encodings:

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See also:

Annotations for §19.28 and Ch.19

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19.28.6 $${\int }_0^1{R}_D(x,y,v^{2}z+(1-v^2)p)dv=R_J(x,y,z,p).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E6

Encodings:

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See also:

Annotations for §19.28 and Ch.19

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19.28.7 $${\int }_0^{\mathrm{\infty }}{R}_J(x,y,z,r^2)dr=\frac{3}{2}\pi R_F(xy,xz,yz),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E7

Encodings:

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See also:

Annotations for §19.28 and Ch.19

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19.28.8 $${\int }_0^{\mathrm{\infty }}{R}_J(tx,y,z,tp)dt=\frac{6}{\sqrt{p}}{R}_C(p,x)R_F(0,y,z).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E8

Encodings:

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See also:

Annotations for §19.28 and Ch.19

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