Whittaker

Whittaker's Functions for the HP-41

Overview

1°) First Kind: M_k,m(x)
2°) Second Kind: W_k,m(x)

1°) First Kind

-Whittaker's function M_k,m(x) is defined by

M_k,m(x) = x^m+1/2 e^-x/2 M(m-k+1/2,2m+1,x)

where M(a,b,x) = Kummer's function.

Data Registers: R00 = x R01-R02: temp
Flags: /
Subroutine: "KUM" ( cf "Kummer's Function for the HP-41" )

01 LBL "WHIM"
02 STO 01
03 RDN
04 ST+ X
05 1
06 +
07 STO 02
08 2
09 /
10 X<>Y
11 -
12 X<> 01
13 XEQ "KUM"
14 RCL 00
15 2
16 /
17 CHS
18 E^X
19 *
20 RCL 00
21 RCL 02
22 2
23 /
24 Y^X
25 *
26 END

( 41 bytes / SIZE 003 )

STACK	INPUTS	OUTPUTS
Z	k	/
Y	m	/
X	x	M_k,m(x)

Example:

2   SQRT
3   SQRT
PI   XEQ "WHIM" >>>> M_{sqrt(2),sqrt(3)}(PI) = 5.612426206              ---Execution time = 12s---

2°) Second Kind

-Whittaker's function of the second kind W_k,m(x) is defined by

W_k,m(x) = [ Gam(-2m) / Gam(-m-k+1/2) ] M_k,m(x) + [ Gam(2m) / Gam(m-k+1/2) ] M_k,-m(x) provided 2m is not an integer.

Data Registers: R00 = x R01-R07: temp
Flags: /
Subroutines: "WHIM" listed above & "GAM" or "GAM+" .... ( cf "Gamma Function for the HP-41" )

01 LBL "WHIW"
02 RDN
03 STO 03
04 X<>Y
05 STO 04
06 X<>Y
07 R^
08 XEQ "WHIM"

09 STO 07
10 RCL 01
11 STO 05
12 RCL 03
13 ST+ X
14 STO 06
15 -
16 XEQ "GAM"

17 ST/ 07
18 RCL 06
19 CHS
20 XEQ "GAM"
21 ST* 07
22 RCL 04
23 RCL 03
24 CHS

25 RCL 00
26 XEQ "WHIM"
27 STO 04
28 RCL 05
29 XEQ "GAM"
30 ST/ 04
31 RCL 06
32 XEQ "GAM"

33 RCL 04
34 *
35 RCL 07
36 +
37 END

( 76 bytes / SIZE 008 )

STACK	INPUTS	OUTPUTS
Z	k	/
Y	m	/
X	x	W_k,m(x)

where 2m is not an integer.

Example:

2   SQRT
3   SQRT
PI   XEQ "WHIW" >>>> W_{sqrt(2),sqrt(3)}(PI) = 2.177593415                 ---Execution time = 34s--

-The last decimal depends on the version of "GAM" that you are using.

References:

[1] Abramowitz and Stegun - "Handbook of Mathematical Functions" - Dover Publications - ISBN 0-486-61272-4
[2] http://functions.wolfram.com/

hp41programs

Whittaker's Functions for the HP-41