Bessel Function J_n(x) for the HP-65

This program is Copyright © 1974 by Hewlett-Packard and is used here by permission. This program was originally published in the HP-65 Math Pac 2.

This program is supplied without representation or warranty of any kind. Hewlett-Packard Company 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.

Card Labels

Bessel Function Jn(x)
Label	Jn(x)
Key	A	B	C	D	E

Overview

This program computes the value of the Bessel function J_n(x) by using a numerical method which makes use of the recurrence relation

          J_n-1(x) = 2n/x * J_n(x) - J_n+1(x)

the summation relation

          J₀(x) + 2 * Σ_{(1=1..infinity)} J_2i(x) = 1

and the fact that

          lim_n→infinityJ_n(x) = 0

First let

          m = INT { 1 + 3x^1/12 + 9x^1/3 + max(n,x)}

where INT means "integer part of''.

Then set

          T_m = a            T_m+1 = 0

where a is an arbitrary non-zero constant.

Then the series of terms, T_k, 0 ≤ k ≤ m, is computed by successively applying the relation

          T_k-1(x) = 2k/x * T_k(x) - T_k+1(x)

starting with k = m.

J_n(x) is then found by dividing the term T_n(x) by the normalizing constant

          K = T₀(x) + 2 Σ_(i=1..p) T_2i(x)

where

          p = m/2    if m is even or
          p - (m-1)/2    if m is odd

Note that all the T_k are proportional to a, hence K and the result are independent of a.

Note: J₀(x) = 1 for x ≤ 10^-6 but it is out of range for this program.

Instructions

Examples

1. J₀(4.7) = -0.27
2. J₅(9.2) = -0.10

The Program

 CODE  KEYS
33 01  STO 1
   43  EEX
   42  CHS
   09  9 
   09  9
33 06  STO 6
   00  0
33 03  STO 3
33 04  STO 4
35 09  g R↑
33 05  STO 5
35 22  g x≤y
   22  GTO
   01  1
   35  g
   04  1/x
   61  +
   35  g
   05  yx
   02  2
   71  x
35 07  g x⇔y
35 22  g x≤y
   44  CLX
   84  R/S

   23  LBL
   01  1
34 01  RCL 1
   06  6
   35  g
   04  1/x
   35  g
   05  yx
   41  ENTER↑
   41  ENTER↑
   09  9
   71  x
   71  x
35 00  g LST X
   31  f
   09  √x
   61  +
   01  1
   61  +
34 01  RCL 1
34 05  RCL 5
35 24  g x>y
35 01  g NOP
35 07  g x⇔y
35 08  g R↓
   61  +
   31  f
   83  INT

   23  LBL
   03  3
33 08  STO 8
34 05  RCL 5
35 23  g x=y
34 06  RCL 6
33 07  STO 7
   00  0
34 08  RCL 8
35 23  g x=y
34 07  RCL 7
   22  GTO
   02  2 
   81  ÷
   32  f-1
   83  INT
35 23  g x=y
34 06  RCL 6
   33  STO
   61  +
   04  4
34 03  RCL 3
34 08  RCL 8
   02  2
34 01  RCL 1
   81  ÷  
   71  x
34 06  RCL 6
33 03  STO 3
   71  x
35 07  g x⇔y
   51  -
33 06  STO 6
34 08  RCL 8
   01  1 
   51  -
   22  GTO 
   03  3

   23  LBL
   02  2
34 04  RCL 4
   02  2
   71  x
34 06  RCL 6
   61  + 
   81  ÷
   84  R/S

Register Use

R1  x
R2
R3  Tk+1
R4  ΣT2i
R5  n
R6  10-99, Tk
R7  Tn
R8  counter k
R9  used

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