Modified Bessel functions

Summary

The modified Bessel functions $I_\nu$ and $K_\nu$ solve the modified Bessel equation $z^2 y'' + z y' - (z^2 + \nu^2) y = 0$. They are the Bessel functions evaluated on the imaginary axis, made real for real arguments. On the positive real axis $I_\nu$ grows like $e^x$ and $K_\nu$ decays like $e^{-x}$. They appear in heat flow, diffusion and electrostatics in cylindrical geometry.