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(f,k) = ó õ f 0   1 Ö 1-k2 sin(q)2 dq

F(50°,sin(40°))=0.91725487

DERIVE calculates this value with PrecisionDigits:=10.

F(50°,sin(40°))=0.9172548679

Now we can use the new Carlson elliptic integral.

RF(x,y,z) = ó õ ¥ 0   1 Ö (x+t)(y+t)(z+t) dt

F(f,k) = sin(f) RF(cos2(f),1-k2 sin2(f),1)

F(50°,sin(40°)) = sin(50°) RF(cos2(50°),1-sin2(40°) sin2(50°),1)

To help DERIVE integrate RF, I will change variables.

RF(x,y,z) = ó õ ¥ 0   t Ö (t2+x) (t2+y) (t2+z) dt

F(50°,sin(40°)) = 0.9172548679

sin(50°) RF(cos2(50°),1-sin2(40°) sin2(50°),1) = 0.9172548679

Carlson has a duplication formulae.

RF(x,y,z) = 2 RF(x+l,y+l,z+l)    where    l = Ö   x y   + Ö   y z   + Ö   z x

In this example x=cos²(f),y=1-k² sin²(f),z=1

x=cos2(50°),y=1-sin2(40°) sin2(50°),z=1

x=0.4131759111,y=0.7575384224,z=1

RF(0.4131759111,0.7575384224,1)=2 RF(2.485791368,2.830153879,3.072615457)

Test the duplication formulae using the example.

RF(0.4131759111,0.7575384224,1)=1.197391190

2 RF(2.485791368,2.830153879,3.072615457)=1.197391190

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

RF(x,x,x)=  sign(x) Öx

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]

64 Ö ave(2856.542188,2856.88655,2857.129012) =1.197391190

F(50°,sin(40°)) =  sin(50°) 64 Ö ave(2856.542188,2856.88655,2857.129012)

F(50°,sin(40°))=0.9172548675

This agrees with A&S page 614.