Bessel Functions

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Bessel function of order is a solution to the differential equation (Bessel equation)

The Bessel functions are named after the German astronomer Friedrich Bessel (1784-1846). These functions first arose when Bessel solved Kepler's equation for describing planetary motion. These functions have been applied in many different physical situation, including the temperature distribution in a circular plate and the shape of a vibrating drumhead.

The Bessel function of order 0 is defined by

Let  .

Then

 

 =

 

             = approaches to for all

 

Thus , by Ratio test , the series converges for all values of .   In other words, the domain of the Bessel function is .

 

Here are the first four partial sums and their graphs :

 

We see that ,  the Bessel function of order 0, is an even function with .   Moreover, by the Alternating Series Estimation Theorem, we have

 

 

 

The animation below shows how the first 25 partial sums approach the Bessel function of order 0.

 

The Bessel function of order 1 is defined by

Let .

Then

 

=                 

 

= approaches to for all          

 

Thus, by Ratio test, the series converges for all values of . In other words, the domain of the Bessel function is also .

 

Here is the graph of where .

 

We see that , the Bessel function of order 1, is an odd function with .

The animation below shows how the first 25 partial sums approach .

 

Here we graph and on a common screen, can you see the relation between them ?

It seems that ,  let's check.

                     =

                                      =

                                                        =

 

To show that satisfies the equation

 

Let , we have

=
             =

　

=
            =

 

=
                     =

Thus,

=

and

= = 0

 

Therefore, .

 

Similarly, we can show that satisfies the equation

 

 

For more information about the Bessel functions and the Bessel equations, please refer to

　

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

　

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