Suppose we expand in a Fourier series:

where

Integrate this by parts to get

The first two terms are zero, so we are left with :

Now, . Hence, since we can also write , we have and

The second integral is zero for integer :

Int(cos(k*M)/k,M=0..Pi): % = value(%);

Thus, we are left with

Now, the Bessel function of the first kind is defined as

where is an integer. Therefore, finally, we have

and

Let's compare this with our previous series expansion example,

Since

,

we know that the series

is absolutely convergent.