In the Sturm-Liouville Boundary Value Problem, there is an important special case called Bessel's Differential Equation which arises in numerous problems, especially in polar and cylindrical coordinates. Bessel's Differential Equation is defined as:
where is a non-negative real number. The solutions of this equation are called Bessel Functions of order . Although the order can be any real number, the scope of this section is limited to non-negative integers, i.e., , unless specified otherwise.
Since Bessel's differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind and the Bessel function of the second kind (also known as the Weber Function) , are needed to form the general solution:
However, is divergent at . The associated coefficient is forced to be zero to obtain a physically meaningful result when there is no source or sink at .