Example 4.25.  The Bessel function of order zero is defined by  

        ,  

and termwise differentiation shows that its derivative is  

             

We leave as an exercise to show that the radius of convergence of these series is infinity.  The Bessel function of order 1 is known to satisfy the differential equation  .

Explore Solution 4.25.

Look at some of the coefficients of and observe the relationship between the coefficients of  .  

We see that   =.  

Aside.  What does the Bessel function    look like ?  Use Mathematica to graph the transformation  .  

The rectangle   was used because the graph of the real function    is decreasing on the interval .  

(c) 2006 John H. Mathews, Russell W. Howell