besselY1¶

Evaluates the real Bessel function of the second kind of order one $Y_1(x)$ .

Synopsis¶

besselY1 (x)

Required Arguments¶

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

Return Value¶

The value of the Bessel function

$Y_1(x) = -\tfrac{1}{\pi} \int_0^{\pi} \sin (\theta - x \sin \theta) d \theta - \tfrac{1}{\pi} \int_0^{\infty} \left\{e^t - e^{-t}\right\} e^{-z \sinh t} dt$

If no solution can be computed, then NaN is returned.

Description¶

This function is sometimes called the Neumann function, $N_1(x)$ , or Weber’s function.

Since $Y_1(x)$ is complex for negative x and is undefined at $x=0$ , besselY1 is defined only for $x>0$ . Because the Bessel function $Y_1(x)$ is oscillatory, its computation becomes inaccurate as x increases.

Example¶

The Bessel function $Y_1(1.5)$ is evaluated.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselY1 import besselY1

x = 1.5
ans = besselY1(x)
print("Y1(%f) = %f" % (x, ans))

Output¶

Y1(1.500000) = -0.412309

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_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 $Y_1(x)$ does not overflow.
`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.