J0

J0 is the Bessel function of the first kind of order zero, denoted J0(x). It is a solution of Bessel’s differential equation x^2 y'' + x y' + x^2 y = 0 that remains finite at the origin. J0 is an entire, even function with J0(0) = 1 and J0'(0) = 0. It is the cylindrical Bessel function of order zero and arises in problems with circular symmetry.

A standard representation is its Maclaurin series J0(x) = sum_{k=0}^∞ (-1)^k (x^2/4)^k / (k!)^2, which converges for all x. Another common property is the derivative relation J0'(x) = -J1(x).

J0 has infinitely many real zeros, with the first positive zero at approximately 2.4048255577. The zeros play a key role in spectral problems and in Fourier–Bessel expansions. For large x, J0(x) has the asymptotic form J0(x) ~ sqrt(2/(π x)) cos(x − π/4).

Integral representations include J0(x) = (1/π) ∫_0^π cos(x sin θ) dθ, and J0(x) = (1/2π) ∫_{-π}^{π} e^{i x sin θ} dθ. Recurrence relations link J0 to other orders, notably J_{n+1}(x) = (2n/x) J_n(x) − J_{n−1}(x).

Applications of J0 span physics and engineering: solving the Helmholtz equation in cylindrical coordinates, vibrations of circular membranes, optical diffraction by circular apertures, antenna theory, and expansions in Fourier–Bessel series.