Bessel function

Bessel functions, named after Friedrich Bessel, are solutions y(x) of "Bessel's differential equation"

x²y'' + xy' + (x² - n²)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.