Appell Hypergeometric Functions for the HP-41
Overview
1°) Appell Hypergeometric Function
F1
2°) Appell Hypergeometric Function
F2
3°) Appell Hypergeometric Function
F3
4°) Appell Hypergeometric Function
F4
-These are functions of 2 variables x , y defined by double series.
1°) Appell Hypergeometric Function F1
-The Appell hypergeometric function F1 is defined by:
F1( a , b , b' , c ; x , y ) = Summ,n=0 to infinity [ (a)m+n (b)m (b')n / (c)m+n ] ( xm / m! ) ( yn /n! )
where (a)k = a(a+1)(a+2).....(a+k-1)
if k > 0 and (a)0 = 1
( Pochhammer symbol )
Data Registers: R00 & R05 to R09: temp ( Registers R01 thru R04 are to be initialized before executing "APLF1" )
• R01 = a • R02 = b
• R04 = c
• R03 = b'
Flags: /
Subroutines: /
| 01 LBL "APLF1"
02 STO 06 03 X<>Y 04 STO 07 05 CLX 06 STO 08 07 STO 09 08 SIGN 09 STO 00 10 SIGN 11 LBL 01 12 LASTX 13 RCL 08 |
14 RCL 09
15 + 16 STO 05 17 RCL 01 18 + 19 * 20 RCL 04 21 RCL 05 22 + 23 / 24 RCL 02 25 RCL 08 26 ST+ Y |
27 ISG X
28 CLX 29 STO 08 30 / 31 * 32 RCL 06 33 * 34 + 35 X#Y? 36 GTO 01 37 CLX 38 STO 08 39 X<> 00 |
40 RCL 01
41 RCL 09 42 + 43 * 44 RCL 03 45 RCL 09 46 + 47 * 48 RCL 04 49 RCL 09 50 ST+ Y 51 ISG X 52 CLX |
53 STO 09
54 * 55 / 56 RCL 07 57 * 58 STO 00 59 + 60 X#Y? 61 GTO 01 62 END |
( 79 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
| Y | y | F1(x,y) |
| X | x | F1(x,y) |
Example: With
a = sqrt(2) b = sqrt(3) b' = sqrt(5)
c = sqrt(6) ( to be stored into
R01 thru R04 )
0.2 ENTER^
0.1 XEQ "APLF1" >>>> F1(
21/2 , 31/2 , 51/2 , 61/2 ,
0.1 , 0.2 ) = 1.489564421
---Execution time = 73s---
Note:
-The series is convergent if Max { | x | , | y | }
< 1
2°) Appell Hypergeometric Function F2
-The Appell hypergeometric function F2 is defined by:
F2( a , b , b' , c , c' ; x , y ) = Summ,n=0
to infinity [ (a)m+n (b)m (b')n
/ (c)m (c')n ] ( xm / m! ) ( yn
/n! )
Data Registers: R00 & R06 to R09: temp ( Registers R01 thru R05 are to be initialized before executing "APLF2" )
• R01 = a • R02 = b
• R04 = c
• R03 = b' • R05 = c'
Flags: /
Subroutines: /
| 01 LBL "APLF2"
02 STO 06 03 X<>Y 04 STO 07 05 CLX 06 STO 08 07 STO 09 08 SIGN 09 STO 00 10 SIGN 11 LBL 01 12 LASTX 13 RCL 01 |
14 RCL 08
15 + 16 RCL 09 17 + 18 * 19 RCL 02 20 RCL 08 21 + 22 * 23 RCL 04 24 RCL 08 25 ST+ Y 26 ISG X |
27 CLX
28 STO 08 29 * 30 / 31 RCL 06 32 * 33 + 34 X#Y? 35 GTO 01 36 CLX 37 STO 08 38 X<> 00 39 RCL 01 |
40 RCL 09
41 + 42 * 43 RCL 03 44 RCL 09 45 + 46 * 47 RCL 05 48 RCL 09 49 ST+ Y 50 ISG X 51 CLX 52 STO 09 |
53 *
54 / 55 RCL 07 56 * 57 STO 00 58 + 59 X#Y? 60 GTO 01 61 END |
( 78 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
| Y | y | F2(x,y) |
| X | x | F2(x,y) |
Example: With
a = sqrt(2) b = sqrt(3) b' = sqrt(5)
c = sqrt(6) c' = sqrt(7)
( to be stored into R01 thru R05 )
0.2 ENTER^
0.1 XEQ "APLF2" >>>> F2(
21/2 , 31/2 , 51/2 , 61/2 ,
71/2 ; 0.1 , 0.2 ) = 1.481853808
---Execution time = 92s---
Note:
-The series is convergent if | x | + | y | <
1
3°) Appell Hypergeometric Function F3
-The Appell hypergeometric function F3 is defined by:
F3( a , a' , b , b' , c ; x , y ) = Summ,n=0
to infinity [ (a)m (a')n (b)m
(b')n / (c)m+n ] ( xm / m! ) ( yn
/n! )
Data Registers: R00 & R06 to R09: temp ( Registers R01 thru R05 are to be initialized before executing "APLF3" )
• R01 = a • R03 = b
• R05 = c
• R02 = a' • R04 = b'
Flags: /
Subroutines: /
| 01 LBL "APLF3"
02 STO 06 03 X<>Y 04 STO 07 05 CLX 06 STO 08 07 STO 09 08 SIGN 09 STO 00 10 SIGN 11 LBL 01 12 LASTX 13 RCL 01 |
14 RCL 08
15 + 16 * 17 RCL 05 18 RCL 08 19 + 20 RCL 09 21 + 22 / 23 RCL 03 24 RCL 08 25 ST+ Y 26 ISG X |
27 CLX
28 STO 08 29 / 30 * 31 RCL 06 32 * 33 + 34 X#Y? 35 GTO 01 36 CLX 37 STO 08 38 X<> 00 39 RCL 02 |
40 RCL 09
41 + 42 * 43 RCL 04 44 RCL 09 45 + 46 * 47 RCL 05 48 RCL 09 49 ST+ Y 50 ISG X 51 CLX 52 STO 09 |
53 *
54 / 55 RCL 07 56 * 57 STO 00 58 + 59 X#Y? 60 GTO 01 61 END |
( 78 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
| Y | y | F3(x,y) |
| X | x | F3(x,y) |
Example: With
a = sqrt(2) a' = sqrt(3) b = sqrt(5)
b' = sqrt(6) c = sqrt(7)
( to be stored into R01 thru R05 )
0.2 ENTER^
0.1 XEQ "APLF3" >>>> F3(
21/2 , 31/2 , 51/2 , 61/2 ,
71/2 ; 0.1 , 0.2 ) = 1.600873822
---Execution time = 62s---
Note:
-The series is convergent if Max { | x | , | y | }
< 1
4°) Appell Hypergeometric Function F4
-The Appell hypergeometric function F4 is defined by:
F4( a , b , c , c' ; x , y ) = Summ,n=0
to infinity [ (a)m+n (b)m+n / (
(c)m (c')n) ] ( xm / m! ) ( yn
/n! )
Data Registers: R00 & R05 to R09: temp ( Registers R01 thru R04 are to be initialized before executing "APLF4" )
• R01 = a • R02 = b
• R03 = c
• R04 = c'
Flags: /
Subroutines: /
| 01 LBL "APLF4"
02 STO 06 03 X<>Y 04 STO 07 05 CLX 06 STO 08 07 STO 09 08 SIGN 09 STO 00 10 SIGN 11 LBL 01 12 LASTX 13 RCL 08 |
14 RCL 09
15 + 16 STO 05 17 RCL 01 18 + 19 * 20 RCL 02 21 RCL 05 22 + 23 * 24 RCL 03 25 RCL 08 26 ST+ Y |
27 ISG X
28 CLX 29 STO 08 30 * 31 / 32 RCL 06 33 * 34 + 35 X#Y? 36 GTO 01 37 CLX 38 STO 08 39 X<> 00 |
40 RCL 01
41 RCL 09 42 + 43 * 44 RCL 02 45 RCL 09 46 + 47 * 48 RCL 04 49 RCL 09 50 ST+ Y 51 ISG X 52 CLX |
53 STO 09
54 * 55 / 56 RCL 07 57 * 58 STO 00 59 + 60 X#Y? 61 GTO 01 62 END |
( 79 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
| Y | y | F4(x,y) |
| X | x | F4(x,y) |
Example: With
a = sqrt(2) b = sqrt(3) c = sqrt(5)
c' = sqrt(6) ( to be stored into
R01 thru R04 )
0.2 ENTER^
0.1 XEQ "APLF4" >>>> F4(
21/2 , 31/2 , 51/2 , 61/2 ,
0.1 , 0.2 ) = 1.490926825
---Execution time = 149s---
Notes:
-The series is convergent if | x |1/2 + | y |1/2 < 1
-All these double series are special cases of "Kampé de Fériet"
functions and may be computed by "KdF"
( cf "Kampé de Fériet Functions for the HP-41"
)
References:
[1] http://mathworld.wolfram.com/AppellHypergeometricFunction.html
[2] http://functions.wolfram.com/
-A treasure-trove! Thousands of formulae that may be downloaded freely!