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:

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

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