Bessel Functions

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:


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

for an arbitrary real or complex number &alpha;. The most common and important special case is where &alpha; is an integer n, then &alpha; is referred to as the order of the Bessel function.

Although &alpha; and &minus;&alpha; produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of &alpha;). Bessel functions are also known as Cylinder functions or Cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates].

The Covenant AI aboard the Ascendant Justice attempted to use Bessel functions to communicate with Cortana.