besselY0¶

Evaluates the real Bessel function of the second kind of order zero $Y_0(x)$ .

Synopsis¶

besselY0 (x)

Required Arguments¶

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

Return Value¶

The value of the Bessel function

$Y_0(x) = \tfrac{1}{\pi} \int_0^{\pi} \sin(x \sin \theta) d \theta - \tfrac{2}{\pi} \int_0^{\infty} e^{-z \sinh \mathrm{t}} dt$

If no solution can be computed, NaN is returned.

Description¶

This function is sometimes called the Neumann function, $N_0(x)$ , or Weber’s function.

Since $Y_0(x)$ is complex for negative x and is undefined at $x=0$ , besselY0 is defined only for $x>0$ . Because the Bessel function $Y_0(x)$ is oscillatory, its computation becomes inaccurate as x increases.

Figure 9.17 — Plot of Y0(x) and Y1(x)

Example¶

The Bessel function $Y_0(1.5)$ is evaluated.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselY0 import besselY0

x = 1.5
ans = besselY0(x)
print("Y0(%f) = %f" % (x, ans))

Output¶

Y0(1.500000) = 0.382449

Warning Errors¶

IMSL_LARGE_ABS_ARG_WARN $|x|$ should be less than $1/\sqrt{\varepsilon}$ where ɛ is the machine precision to prevent the answer from being less accurate than half precision.

Fatal Errors¶

IMSL_LARGE_ABS_ARG_FATAL $|x|$ should be less than $1/\varepsilon$ where ɛ is the machine precision for the answer to have any precision.