Bessel Function Calculator

Bessel functions are solutions to Bessel’s differential equation: x^2 * y’’ + x * y’ + (x^2 - n^2) * y = 0

They appear whenever a problem has cylindrical symmetry — vibrations of a circular drumhead, heat conduction in a cylinder, electromagnetic waves in a coaxial cable, water waves on a circular pond, signal processing for FM synthesis. Anywhere polar coordinates show up in physics, Bessel functions tend to follow.

Two families exist for each integer order n:

J_n (Bessel function of the first kind) is finite at the origin and oscillates with slowly decaying amplitude as x grows. J_0 starts at 1 when x=0, dips to about -0.4 around x=4, and continues oscillating.

Y_n (Bessel function of the second kind, also called Neumann function) goes to negative infinity as x approaches 0, and oscillates similarly to J_n for large x. Y functions are only defined for x greater than 0.

Both are linearly independent, so the general solution to Bessel’s equation of order n is c1 * J_n(x) + c2 * Y_n(x).

This calculator returns J0, J1, Y0, and Y1 simultaneously — the four most commonly used. The polynomial approximations are from Abramowitz and Stegun, accurate to about 10^-7 for all positive x. Below x=8 it uses rational polynomial fits; above that, an asymptotic form involving sine and cosine.

Worked check: J_0(0) = 1 exactly. J_0(2.4048) is approximately 0 — the first zero of J_0, important in determining the lowest vibration frequency of a circular drumhead. J_1(0) = 0 (J_1 starts at zero by symmetry).