besselK1¶

Evaluates the real modified Bessel function of the second kind of order one $K_1(x)$ .

Synopsis¶

besselK1 (x)

Required Arguments¶

float x (Input): Point at which the Bessel function is to be evaluated. It must be positive.

Return Value¶

The value of the Bessel function

$K_1(x) = \int_0^{\infty} \sin (x \sinh t) \sinh t\phantom{.}dt$

If no solution can be computed, NaN is returned.

Description¶

Since $K_1(x)$ is complex for negative x and is undefined at $x=0$ , besselK1 is defined only for $x>0$ . For large x, besselK1 will underflow. See Figure 9-12 for a graph of $K_1(x)$ .

Example¶

The Bessel function $K_1(1.5)$ is evaluated.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselK1 import besselK1

x = 1.5
ans = besselK1(x)
print("K1(%f) = %f" % (x, ans))

Output¶

K1(1.500000) = 0.277388

Alert Errors¶

IMSL_LARGE_ARG_UNDERFLOW

The argument x must not be so large that the result, approximately equal to $\sqrt{\pi/(2x)} e^{-x}$ ,

underflows.

Fatal Errors¶

IMSL_SMALL_ARG_OVERFLOW The argument x must be large enough ( $x>\max(1/b,s)$ where s is the smallest representable positive number and b is the largest representable number) that $K_1(x)$ does not overflow.