besselExpI0

Evaluates the exponentially scaled modified Bessel function of the first kind of order zero.

Synopsis

besselExpI0 (x)

Required Arguments

float x (Input)

Point at which the Bessel function is to be evaluated.

Return Value

The value of the scaled Bessel function \(e^{-|x|} I_0(x)\). If no solution can be computed, NaN is returned.

Description

The Bessel function \(I_0(x)\) is defined to be

\[I_0(x) = \tfrac{1}{\pi} \int_0^{\pi} \cosh (x \cos \theta) d \theta\]

Example

The expression \(e^{-4.5} I_0(4.5)\) is computed directly by calling besselExpI0 and indirectly by calling besselI0. The absolute difference is printed. For large x, the internal scaling provided by besselExpI0 avoids overflow that may occur in besselI0.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselExpI0 import besselExpI0
from pyimsl.math.besselI0 import besselI0

x = 4.5
ans = besselExpI0(x)
print("(e**(-4.5))I0(4.5) = %f" % (ans))
error = abs(ans - (exp(-x) * besselI0(x)))
print("Error = %e" % (error))

Output

(e**(-4.5))I0(4.5) = 0.194198
Error = 2.775558e-17