hp41programs

Lobachevsky

Lobachevsky Function for the HP-41


Overview
 

-The Lobachevsky function is defined by  L(µ) = - §0µ  Ln | 2 Sin t | dt

-We have also   L(µ) = (1/2) Im [ Li2 ( exp(2i.µ) ) ]    where   Li2 = dilogarithm function

-So, we could call "DILM1" that is listed in "Dilogarithm & Polylogarithm functions for the HP-41" , paragraph 3°)
-The following routine employs the same method with a few extra terms.
 

  L(µ) = (1/2) SUMk=1,2,.....  sin(2k.µ) / k2 = -2.µ ln | 2.µ | + 2.µ + SUMk=1,2,.....  (-1)k-1/(2k+1)! [B2k/(2k)] (2.µ)2k+1

  where  B2k  are the Bernoulli numbers
 

Program Listing
 
 

Data Registers:  R00 = 2 µ ( in radians )
Flags: /
Subroutines: /
 
 

 01  LBL "LOB"
 02  ST+ X
 03  FC? 43
 04  D-R
 05  PI
 06  ST+ X
 07  MOD
 08  PI
 09  X>Y?
 10  CLX
 11  ST+ X
 12  -
 13  STO 00       
 14  X^2
 15  918 E17
 16  1/X
 17  RCL Y		  18  43 E20
 19  /
 20  +
 21  *
 22  1931 E15
 23  1/X
 24  +
 25  *
 26  3983 E13
 27  1/X
 28  +
 29  *
 30  12462 E-19
 31  +
 32  *
 33  15
 34  FACT         		  35  12
 36  *
 37  1/X
 38  +
 39  *
 40  29522 E7
 41  1/X
 42  +
 43  *
 44  11
 45  FACT         
 46  132
 47  *
 48  1/X
 49  +
 50  *
 51  9		  52  FACT
 53  240
 54  *
 55  1/X
 56  +
 57  *
 58  7
 59  FACT         
 60  252
 61  *
 62  1/X
 63  +
 64  *
 65  5
 66  FACT
 67  X^2
 68  1/X		  69  +
 70  *
 71  72
 72  1/X
 73  +
 74  *
 75  RCL 00        
 76  ABS
 77  X#0?
 78  LN
 79  -
 80  RCL 00
 81  ST* Y
 82  +
 83  2
 84  /
 85  END

 
    ( 144 bytes / SIZE 001 )
 
 

      STACK        INPUT      OUTPUT
           X             µ          L(µ)

  where µ must be expressed in degrees if the HP-41 is in DEG mode, or in radians if the calculator is in RAD mode.

Example:

   XEQ "DEG"

   28°90854705  XEQ "LOB"  >>>>  L(28°90854705) = 0.5071518525

Notes:

-If your HP-41 is in GRAD mode, express µ in degrees.

-"LOB" is a periodic function of period PI ( or 180° )
-Its graph looks approximately like this:

      y
       |               *                      y = L(x)
       |     *                   *
       | *                                *
   0  |*---------------------------*----------------------------*PI------- x (rd)
       |                                                 *                                  *
       |                                                          *                     *
       |                                                                      *