*x*^{2}*y*'' +*xy*' + (*x*^{2}-*n*^{2})*y*= 0

for non-negative integer values of *n*.

They come in two kinds:

- Bessel functions of the first kind
*J*(_{n}*x*), the solutions of the above differential equation which are defined for*x*= 0. - Bessel functions of the second kind
*Y*(_{n}*x*), the solutions which are non-singular (infinite) for*x*= 0.

The graphs of Bessel function look like oscillating sine or cosine functions which "level off" because they have been divided by a term of the order of √*x*.

They are important in many physical problems including those involving spherical or cylindrical? coordinates, and in frequency modulation.

**Applications:**

- the solution to Schrödinger's equation for the Hydrogen atom.