`besseli`

Modified Bessel function of the first kind.

Syntax

I = besseli(nu, Z)

Returns the modified Bessel function I_nu(Z).

I = besseli(nu, Z, 1)

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

Description

I_nu solves z^2*w'' + z*w' - (z^2 + nu^2)*w = 0. It is the Bessel equation with the sign of the z^2 term flipped, so instead of oscillating the solution grows exponentially. I_0(0) is 1, and I_nu(0) is 0 for nu > 0.

The scaled form is useful when Z is large and the plain value would overflow.

Input arguments

nu: Order (real).
Z: Real or complex argument.
scale: Optional 1 to scale the result and avoid overflow.

Output arguments

I: Values of the function, same size as Z (after expanding nu).

Example

Modified Bessel function

besseli(0, 1)
besseli(1, 1, 1)

Output

ans = 1.2661
ans = 0.2079

besseli