Bessel functions are both of the canonical solutions to Bessel's homogeneous differential equation:

These equations arise naturally in the solution of Laplace's equation in cylindrical coordinates. The solutions of the non-homogeneous Bessel differential equation shown in the following diagram are called Struve functions:

In either case, the order of the equation is the complex number alpha which acts as a parameter. Depending on the canonical solution and the order, the Bessel and Struve functions are addressed (and computed) differently.

For Bessel functions, we have algorithms to produce Bessel functions of the first kind (jv) and second kind (yn and yv), Hankel functions of the first and second kind (hankel1 and hankel2), and the modified Bessel functions of the first and second kind (iv, kn, and kv). Their syntax is similar in all cases: first parameter is the order and second parameter the independent variable. The component n in the definition indicates that an integer is to be used as the order (since they are optimally coded for that situation):

>>> import numpy
>>> import scipy.special
>>> scipy.special.jn(5,numpy.pi)

The output is as follows:

0.052141184367118461

The scipy.special module also contains fast versions of the most common Bessel functions (those of orders 0 and 1): j0(x), j1(x) (first kind y0(x)and second kind y1(x)), and so on. There are definitions of the spherical Bessel functions, such as sph_jn(n,z) and sph_yn(z); the Riccati-Bessel functions, such as riccati_jn(n,x) and riccati_yn(n,x); and derivatives of all the basic ones, such as jvp, yvp, kvp, ivp, h1vp, and h2vp.

For Struve functions, we have fast algorithms to compute solutions of the differential equation of order v:(struve(v,x) and modstruve(v,x)).