# Bessel's Equation - HP 49g+ User Manual

Graphing calculator.

## Bessel's equation

The ordinary differential equation x
where the parameter ν is a nonnegative real number, is known as Bessel's
differential equation.
Bessel functions of the first kind of order ν:
J
(
x
)
ν
where ν is not an integer, and the function Gamma Γ(α) is defined in Chapter
3.
If ν = n, an integer, the Bessel functions of the first kind for n = integer are
defined by
J
(
n
Regardless of whether we use ν (non-integer) or n (integer) in the calculator,
we can define the Bessel functions of the first kind by using the following finite
series:
Thus, we have control over the function's order, n, and of the number of
elements in the series, k.
function DEFINE to define function J(x,n,k).
series, calculate J(0.1,3,5), i.e., in RPN mode: .1#3#5@@@J@@@
The result is 2.08203157E-5.
2
⋅(d
2
2
y/dx
) + x⋅ (dy/dx)+ (x
Solutions to Bessel's equation are given in terms of
m
(
) 1
ν
x
2
m
ν
2
m
!
m
0
m
(
) 1
n
x
)
x
2
m
n
2
m
( !
m
0
Once you have typed this function, you can use
This will create the variable @@@J@@@
For example, to evaluate J
2
2
) ⋅y = 0,
2
m
x
,
(
ν
m
) 1
2
m
x
.
n
m
)!
(0.1) using 5 terms in the
3
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