TEX source
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(ϕ,k) = ⌠
⌡ 		ϕ

0 
 1

√
1−k2 sin(θ)2
 dθ

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) = ⌠
⌡ 		∞

0 
 1

√
(x+t)(y+t)(z+t)
 dt

F(ϕ,k) = sin(ϕ) RF(cos2(ϕ),1−k2 sin2(ϕ),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) = ⌠
⌡ 		∞

0 
 t

√
(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+λ,y+λ,z+λ)    where    λ =
√
 
x y
 
 +
√
 
y z
 
 +
√
 
z x
 
In this example x=cos2(ϕ),y=1−k2 sin2(ϕ),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)
√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]

64

√
ave(2856.542188,2856.88655,2857.129012)
 =1.197391190

F(50°,sin(40°)) = sin(50°) 64

√
ave(2856.542188,2856.88655,2857.129012)

F(50°,sin(40°))=0.9172548675
This agrees with A&S page 614.

File translated from TEX by TTH, version 3.88.
On 28 Aug 2010, 06:59.