The Museum of HP Calculators

# Kummer's Function for the HP-41

This program is Copyright © 2006 by Jean-Marc Baillard and is used here by permission.

This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

Overview

1°) Real Variable
2°) Complex Variable

-Kummer's function  M(a,b,x) is defined by

M(a;b;x) = 1 +  (a)1/(b)1. x1/1! + ............. +  (a)n/(b)n . xn/n! + ..........  where (a)n = a(a+1)(a+2) ...... (a+n-1)

1°) Real Variable

Data Registers:             R00 = x
•   R01 = a
•   R02 = b              registers R01 R02 are to be initialized before executing "KUM"
Flags: /
Subroutines:  /

01  LBL "KUM"
02  STO 00
03  CLST
04  SIGN
05  ENTER^
06  STO T
07  LBL 01
08  X<> T
09  RCL 01
10  R^
11  ST+ Y
12  RDN
13  *
14  RCL 02
15  R^
16  ST+ Y
17  ISG X
18  CLX
19  ST* Y
20  RDN
21  /
22  RCL 00
23  *
24  STO T
25  X<>Y
26  ST+ Y
27  X#Y?
28  GTO 01
29  END

( 46 bytes / SIZE 002 )

 STACK INPUTS OUTPUTS X x M(a;b;x) L / x

Example:     Compute  M(2;3;-Pi)

2  STO 01
3  STO 02
PI   CHS   XEQ "KUM"  yields  0.166374562   ( in 13 seconds )

Notes:

a)   2x (Pi)-1/2 M(1/2;3/2;-x2) = erf(x) = error function
b)        (x/2)n  M(n+1/2;2n+1;2x) = Gamma(1+n) ex In(x)        where     In  = a modified Bessel function
c)       (xa/a)   M(a;a+1;-x)  =  incgam(a;x)  = §0x e-t ta-1 dt                ( incgam = incomplete gamma function )

and many other functions are related to Kummer's functions.

2°) Complex Variable

-The parameters  a & b are still real, but the variable  z = x + i.y  is complex

Data Registers:        R00 and R03 thru R08: temp
•   R01 = a
•   R02 = b              registers R01 R02 are to be initialized before executing "KUMZ"
Flags: /
Subroutines:  /

01  LBL "KUMZ"
02  R-P
03  STO 00
04  X<>Y
05  STO 03
06  CLX
07  STO 05
08  STO 06
09  STO 08
10  SIGN
11  STO 04
12  STO 07
13  LBL 01
14  RCL 03
15  RCL 08
16  +
17  STO 08
18  RCL 01
19  RCL 02
20  RCL 06
21  ST+ Z
22  +
23  ISG 06
24  CLX
25  RCL 06
26  *
27  /
28  RCL 00
29  *
30  RCL 07
31  *
32  STO 07
33  P-R
34  RCL 04
35  +
36  STO04
37  LASTX
38  -
39  X^2
40  X<>Y
41  RCL 05
42  +
43  STO 05
44  LASTX
45  -
46  X^2
47  +
48  X#0?
49  GTO 01
50  RCL 05
51  RCL 04
52  END

( 64 bytes / SIZE 009 )

 STACK INPUTS OUTPUTS Y y y' X x x'

with  Kum ( a ; b ; x+i.y ) = x' + i.y'

Example:     If  a = 4 ;  b = 3     4  STO 01   3  STO 02

2  ENTER^
1  XEQ "KUMZ  >>>>   -3.156090293   X<>Y   2.541499313

Whence    Kum ( 4 ; 3 ; 1 + 2.i ) =  -3.156090293 + i. 2.541499313

Reference:       Abramowitz and Stegun , "Handbook of Mathematical Functions" -  Dover Publications -  ISBN  0-486-61272-4