DERIVE source

This is an example of how the new Carlson elliptic integral works. This was written by Jim FitzSimons cherry@neta.com. From A&S page 614

$F(\phi ,k)={\int }_0^{\phi }\frac{1}{\sqrt{1-k^2\hspace{1em}\sin (\theta ){}^2}}d\theta$

$F({50}^{ˆ},\sin ({40}^{ˆ}))=0.91725487$

DERIVE calculates this value with PrecisionDigits:=10.

$F({50}^{ˆ},\sin ({40}^{ˆ}))=0.9172548679$

Now we can use the new Carlson elliptic integral.

$\mathrm{RF}(x,y,z)={\int }_0^{\infty }\frac{1}{\sqrt{(x+t)(y+t)(z+t)}}\mathrm{dt}$

$F(\phi ,k)=\sin (\phi )\hspace{1em}\mathrm{RF}(\cos {}^2(\phi ),1-k^2\hspace{1em}\sin {}^2(\phi ),1)$

$F({50}^{ˆ},\sin ({40}^{ˆ}))=\sin ({50}^{ˆ})\hspace{1em}\mathrm{RF}(\cos {}^2({50}^{ˆ}),1-\sin {}^2({40}^{ˆ})\hspace{1em}\sin {}^2({50}^{ˆ}),1)$

To help DERIVE integrate RF, I will change variables.

$\mathrm{RF}(x,y,z)={\int }_0^{\infty }\frac{t}{\sqrt{(t^2+x)\hspace{1em}(t^2+y)\hspace{1em}(t^2+z)}}\mathrm{dt}$

$F({50}^{ˆ},\sin ({40}^{ˆ}))=0.9172548679$

$\sin ({50}^{ˆ})\hspace{1em}\mathrm{RF}(\cos {}^2({50}^{ˆ}),1-\sin {}^2({40}^{ˆ})\hspace{1em}\sin {}^2({50}^{ˆ}),1)=0.9172548679$

Carlson has a duplication formulae.

$\mathrm{RF}(x,y,z)=2\hspace{1em}\mathrm{RF}(x+\lambda ,y+\lambda ,z+\lambda )\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{where}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\lambda =\sqrt{x\hspace{1em}y}+\sqrt{y\hspace{1em}z}+\sqrt{z\hspace{1em}x}$

In this example $x=\cos {}^2(\phi ),y=1-k^2\hspace{1em}\sin {}^2(\phi ),z=1$

$x=\cos {}^2({50}^{ˆ}),y=1-\sin {}^2({40}^{ˆ})\hspace{1em}\sin {}^2({50}^{ˆ}),z=1$

$x=0.4131759111,y=0.7575384224,z=1$

$\mathrm{RF}(0.4131759111,0.7575384224,1)=2\hspace{1em}\mathrm{RF}(2.485791368,2.830153879,3.072615457)$

Test the duplication formulae using the example.

$\mathrm{RF}(0.4131759111,0.7575384224,1)=1.197391190$

$2\hspace{1em}\mathrm{RF}(2.485791368,2.830153879,3.072615457)=1.197391190$

When x, y, and z are equal RF simplifies.

$\mathrm{RF}(x,x,x)=\frac{\mathrm{sign}(x)}{\sqrt{x}}$

As we apply the duplication formulae the values of $x$, $y$, and $z$ become equal.

$[0.4131759111,0.7575384224,1]$

$[2.485791368,2.830153879,3.072615457]$

$[10.85074668,11.19510919,11.43757077]$

$[44.32831583,44.67267834,44.91513992]$

$[178.2429880,178.5873505,178.8298121]$

$[713.9027732,714.2471357,714.4895973]$

$[2856.542188,2856.886550,2857.129012]$

$\frac{64}{\sqrt{\mathrm{ave}(2856.542188,2856.88655,2857.129012)}}=1.197391190$

$F({50}^{ˆ},\sin ({40}^{ˆ}))=\frac{\sin ({50}^{ˆ})\hspace{1em}64}{\sqrt{\mathrm{ave}(2856.542188,2856.88655,2857.129012)}}$

$F({50}^{ˆ},\sin ({40}^{ˆ}))=0.9172548675$

This agrees with A&S page 614.