Bessel function of the first kind

Summary

The Bessel function of the first kind $J_\nu(z)$ is the solution of Bessel's equation $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$ that behaves like $\frac{1}{\Gamma(\nu + 1)} \left( \frac z2 \right)^\nu$ near $z = 0$. This description assumes that $\nu$ is not a negative integer. It is given by a power series that converges for every $z$. Bessel functions describe waves and heat flow in cylinders and discs, such as the vibration modes of a drum. They appear throughout analysis, number theory and probability.