Bessel functions, are canonical solutions of Bessel’s differential equation:
for an arbitrary real or complex number (the order of the Bessel function). The most common and important special case is where is an integer (in which case we call it ).
Bessel functions of the first kind, denoted , are solutions of Bessel’s differential equation that are finite at the origin () for non-negative integer , and diverge as for negative non-integer . It is possible to define the function by its Taylor series expansion around :
where is the gamma function, a generalization of the factorial function to non-integer values.
Bessel functions of the second kind, denoted by , are solutions of the Bessel differential equation. They are singular (infinite) at the origin ().
For non-integer , it is related to by:
In the case of integer order , the function is defined by taking the limit as a non-integer tends to :
which has the result (in integral form)