besselJ1

Evaluates the real Bessel function of the first kind of order one \(J_1(x)\).

Synopsis

besselJ1 (x)

Required Arguments

float x (Input)

Point at which the Bessel function is to be evaluated.

Return Value

The value of the Bessel function

\[J_1(x) = \frac{1}{\pi} \int_0^{\pi} \cos (x \sin \theta - \theta) d \theta\]

If no solution can be computed, NaN is returned.

Description

Because the Bessel function \(J_1(x)\) is oscillatory, its computation becomes inaccurate as \(|x|\) increases.

Example

The Bessel function \(J_1(1.5)\) is evaluated.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselJ1 import besselJ1

x = 1.5
ans = besselJ1(x)
print("J1(%f) = %f" % (x, ans))

Output

J1(1.500000) = 0.557937

Alert Errors

IMSL_SMALL_ABS_ARG_UNDERFLOW

To prevent \(J_1(x)\) from underflowing, either x must be zero, or \(|x|>2s\) where s is the smallest representable positive number.

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.