Bessel function

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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 Jn(x), the solutions of the above differential equation which are defined for x = 0.
  • Bessel functions of the second kind Yn(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.