Week 13-14

Bessel functions:

Bessel functions, first defined by the mathematician Daneil Bernoulli and then generalized by Fridrich Bessel are canonical solutions y(x) of Bessel's differential equation.

{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}

for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.