`besselj`

Bessel function of the first kind.

Syntax

J = besselj(nu, Z)

Returns J_nu(Z).

J = besselj(nu, Z, 1)

Returns the function scaled by exp(-abs(imag(Z))).

Description

J_nu solves Bessel's equation z^2*w'' + z*w' + (z^2 - nu^2)*w = 0 and is the solution that stays finite at the origin. It looks like a decaying cosine or sine: it oscillates with an amplitude shrinking like 1/sqrt(z). These functions appear in the vibration of circular membranes, waveguides and diffraction.

If nu and Z are arrays of compatible shapes, the result has the broadcast size; a row nu with a scalar Z gives one value per order.

Input arguments

nu: Order (real, not necessarily an integer).
Z: Real or complex argument.

Output arguments

J: Values of the function. For real Z >= 0 the values are real.

Example

Orders 0, 1 and 2 at x = 1

besselj(0:2, 1)
besselj(1, [0 1 2])

Output

ans = 1×3

    0.7652    0.4401    0.1149
ans = 1×3

         0    0.4401    0.5767

besselj