Theta

# Theta Functions for the HP-41

Overview

Definitions:       with  q =  e-pi K'/K  ( see Jacobian elliptic functions for the HP-41 )

Theta1(x;q) =  2.q1/4 SUMk>=0  (-1)k qk(k+1) sin(2k+1)x
Theta2(x;q) =  2.q1/4 SUMk>=0    qk(k+1) cos(2k+1)x
Theta3(x;q) =  1 + 2 SUMk>=1    qk*k cos 2kx                                  ( -1 < q < 1 )
Theta4(x;q) =  1 + 2 SUMk>=1  (-1)k qk*k cos 2kx

Program Listing

Data Registers:  R00-R03: temp  R01 = x  R02 = q
Flags: F01 , F02 , F03 , F04   Set  only one of these 4 Flags
Subroutines: /

-If flag F01 is set , "THETA" calculates Theta1(x;q)
-If flag F02 is set , "THETA" calculates Theta2(x;q)
-If flag F03 is set , "THETA" calculates Theta3(x;q)
-If flag F04 is set , "THETA" calculates Theta4(x;q)

-Synthetic registers M , N , O may be replaced by any unused data registers.

 01 LBL "THETA"  02 STO 01  03 10  04 CHS  05 RCL Z  06 STO 02  07 ABS  08 X#0?  09 LOG  10 X#0?  11 /  12 SQRT  13 INT  14 STO 00  15 1  16 FC? 01 17 FS? 04  18 CHS  19 X<>Y  20 Y^X  21 STO 03            22 CLX  23 RAD  24 ISG 00  25 LBL 01  26 RCL 02  27 RCL 00  28 ENTER  29 ST* Y  30 FC? 03  31 FS? 04  32 CLX 33 +  34 Y^X  35 RCL 03  36 FC? 01  37 FS? 04  38 CHS  39 STO 03            40 *  41 RCL 00  42 ST+ X  43 RCL 01  44 ST* Y  45 FC? 03  46 FS? 04  47 CLX  48 + 49 FS? 01  50 SIN  51 FC? 01  52 COS  53 *  54 +  55 DSE 00  56 GTO 01  57 FC? 01  58 FS? 02  59 RCL 01            60 FS? 01  61 SIN  62 FS? 02  63 COS 64 DEG  65 FC? 03  66 FS? 04  67 .5  68 +  69 ST+ X  70 RCL 02            71 ABS  72 SQRT  73 SQRT  74 FC? 03  75 FS? 04  76 SIGN  77 *  78 END

( 116 bytes / SIZE 004 )

 STACK INPUTS OUTPUTS Y q / X x Thetan (x,q)

Example:   Compute  Theta1(x,q) , Theta2(x,q) , Theta3(x,q) , Theta4(x,q)   for  x = 2 , q = 0.3

- CF02 CF 03 CF 04

- SF 01                0.3  ENTER^   2   XEQ "THETA"  gives      1.382545289             ---Execution time = 10s---
- CF01   SF 02    0.3  ENTER^   2         R/S              gives     -0.488962527
- CF 02  SF 03    0.3  ENTER^   2         R/S              gives      0.605489938                           idem
- CF 03  SF 04    0.3  ENTER^   2         R/S              gives      1.389795845

Reference:

[1]  Abramowitz and Stegun , "Handbook of Mathematical Functions" -  Dover Publications -  ISBN  0-486-61272-4