Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)

[Gerson W. Barbosa]

001:LBL  A ; Hypergeometric function –> F(a, b; c; z)
002:STOS  01
003:x⇆  Y 
004:/
005:×
006:×
007:RCL  X
008:1
009:STO+  Y
010:x⇆  T 
011:⇆  ZTYX
012:INC  04
013:INC  03
014:INC  02
015:INC  Y
016:RCL×  04
017:RCL/  02
018:RCL×  03
019:RCL/  Y
020:RCL×  01
021:RCL  Z
022:R↓
023:STO+  Z
024:⇆  TZXY
025:x≠?  Y
026:BACK  015
027:RTN

028:LBL  B ; Gauss-Kummer –> P = π(a + b)F(-1/2, -1/2; 1; h²)
029:©CONJ ; where h = (a - b)/(a + b)
030:©RCL  L
031:x⇆  Y 
032:©+
033:STO  I
034:/
035:x²
036:#  1/2L
037:+/-
038:RCL  X
039:#  001
040:R↑
041:XEQ  A
042:RCL×  I
043:#  π  
044:×  π  
045:RTN

046:LBL  C ; Approximation formula using AGM
047:©ENTER; P ~ 2π{a + b - a*b/AGM(a,b) - 2[AGM(a,b) - √(a*b)]}
048:STO+  T
049:RCL×  Y
050:√
051:STO  I
052:x⇆  L
053:⇆  ZYXT
054:AGM
055:STO/  Y
056:RCL-  I
057:STO+  X
058:+
059:-
060:#  π  
061:×  
062:STO+  X
063:END


Examples:

8 ENTER 9 B -> 53.45328500297187553380768922447455

8 ENTER 9 C -> 53.45328500297(5636)


2 ENTER 3 B -> 15.86543958929058979133166302778306

2 ENTER 3 C -> 15.865439(6104)

Note: if a/b > 3, where a is the semi-major axis, then Cayley’s method (not implemented here) is more efficient. Please refer to this paper for more details:

http://web.tecnico.ulisboa.pt/~mcasquilh...llipse.pdf

——

A couple of hypergeometric function identities:

ln(1 + x) = xF(1, 1; 2; -x)

arcsin(x) = xF(1/2, 1/2; 3/2; x²)

Examples:

1 ENTER ENTER 2 ENTER 0.5 A 0.5 +/- * -> -0.69314718056
2 LN + -> 6e-34

0.5 ENTER ENTER 1.5 ENTER 0.25 A 0.5 * -> 0.5235987756
6 * π - -> 1e-32