This program is Copyright © 1974 by HewlettPackard and is used here by permission. This program was originally published in the HP65 Math Pac 2.
This program is supplied without representation or warranty of any kind. HewlettPackard 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 J_{n}(x) 

Label 
J_{n}(x) 




Key 
A 
B 
C 
D 
E 
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_{n1}(x) = 2n/x * J_{n}(x)  J_{n+1}(x)
the summation relation
J_{0}(x) + 2 * _{(1=1..infinity) }J_{2i}(x) = 1
and the fact that
lim_{n>infinity}J_{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 nonzero constant.
Then the series of terms, T_{k}, 0 <= k <= m, is computed by successively applying the relation
T_{k1}(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 = T_{0}(x) + 2 _{(i=1..p)} T_{2i}(x)
where
p = m/2 if m is even or
p  (m1)/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: J_{0}(x) = 1 for x <= 10^{6} but it is out of range for this program.
Step 
Instructions 
Input Data/Units 
Keys 
Output Data/Units 
1 
Enter Program 

2 

n 
ENTER 

3 

x 
A 
J_{n}(x) 
1. J_{0}(4.7) = 0.27
2. J_{5}(9.2) = 0.10
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 y^{x} 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 y^{x} 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 T_{k+1} R4 T_{2i} R5 n R6 10^{99}, T_{k} R7 T_{n} R8 counter k R9 used
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