Bessel functions
/ BESS-ul /
Just as the circle gives us sine and cosine, the cylinder gives us Bessel functions — the special functions that solve the Bessel equation. They look like sine and cosine that have been gently stretched and shrunk: they oscillate forever, crossing zero again and again, but their amplitude slowly fades and the spacing of their zeros slowly drifts. They are the natural vocabulary for anything that ripples outward in a circle.
There are two main kinds for each order nu. The Bessel function of the first kind, J_nu(x), is the well-behaved solution: it is finite at the centre x = 0 (for nu >= 0) and is the one you keep in physical problems on a solid disk or cylinder. It is given by the Frobenius series J_nu(x) = sum from m=0 of (-1)^m / (m! Gamma(m + nu + 1)) times (x/2)^(2m + nu), where the Gamma function supplies the factorials for non-integer order. The Bessel function of the second kind, Y_nu(x) (also called the Neumann function), is the partner that blows up — it goes to minus infinity like a logarithm at x = 0 — and is needed for problems on a region that excludes the centre, such as the annulus between two pipes. For integer order it is precisely Y_n that carries the logarithm demanded by the equal-root Frobenius case.
Their indispensable properties are oscillation, an infinite ladder of zeros (which set the allowed frequencies in drum and waveguide problems), orthogonality over a disk with weight x (the basis of Fourier-Bessel series), and tidy recurrences that link J_(nu-1), J_nu, J_(nu+1) and their derivatives. For large x they settle into the clean asymptotic form J_nu(x) is approximately sqrt(2/(pi x)) cos(x - nu pi/2 - pi/4), making the 'stretched sine' picture exact in the limit.
For the half-integer order nu = 1/2 the Bessel function is elementary: J_(1/2)(x) = sqrt(2/(pi x)) sin(x). This is the rare case where a Bessel function reduces to familiar functions — it is exactly the 'stretched sine' picture made literal, with amplitude shrinking like 1/sqrt(x).
Half-integer orders give elementary Bessel functions; integer and general orders do not, which is why the functions earn their own name and tables.
J_nu and J_(-nu) are independent solutions only when nu is not an integer; for integer n, J_(-n) = (-1)^n J_n is not new, and you must bring in Y_n (with its logarithm) for the genuine second solution.