The Museum of HP Calculators

# Bessel Function Jn(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.

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

## Overview

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

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

the summation relation

J0(x) + 2 * (1=1..infinity) J2i(x) = 1

and the fact that

limn->infinityJn(x) = 0

First let

m = INT { 1 + 3x1/12 + 9x1/3 + max(n,x)}

where INT means "integer part of''.

Then set

Tm = a            Tm+1 = 0

where a is an arbitrary non-zero constant.

Then the series of terms, Tk, 0 <= k <= m, is computed by successively applying the relation

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

starting with k = m.

Jn(x) is then found by dividing the term Tn(x) by the normalizing constant

K = T0(x) + 2 (i=1..p) T2i(x)

where

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

Note that all the Tk 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

 Step Instructions Input Data/Units Keys Output Data/Units 1 Enter Program 2 n ENTER 3 x A Jn(x)

## 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 roll up
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  sqrt
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 roll dn
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
```

```R1  x
R2
R3  Tk+1
R4  T2i
R5  n
R6  10-99, Tk
R7  Tn
R8  counter k
R9  used
```