besselExpK0

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

Synopsis

besselExpK0 (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 \(e^x K_0(x)\). If no solution can be computed, NaN is returned.

Description

The argument must be greater than zero for the result to be defined. The Bessel function \(K_0(x)\) is defined to be

\[K_0(x) = \int_0^{\infty} \cos (x \sinh t) dt\]

Example

The expression

\[\sqrt{e} K_0 (0.5)\]

is computed directly by calling besselExpK0 and indirectly by calling besselK0. The absolute difference is printed. For large x, the internal scaling provided by besselExpK0 avoids underflow that may occur in besselK0.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselExpK0 import besselExpK0
from pyimsl.math.besselK0 import besselK0

x = 0.5
ans = besselExpK0(x)
print("(e**(0.5))K0(0.5) = %f" % (ans))

error = abs(ans - (exp(-x) * besselK0(x)))
print("Error = %e" % (error))

Output

(e**(0.5))K0(0.5) = 1.524109
Error = 9.634209e-01