naginterfaces.library.specfun.bessel_​j0_​real

naginterfaces.library.specfun.bessel_j0_real(x)

bessel_j0_real returns the value of the Bessel function $J_0\left(x\right)$.

For full information please refer to the NAG Library document for s17ae

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s17aef.html

Parameters

xfloat

The argument $x$ of the function.

Returns

j0float

The value of the Bessel function $J_0\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $\left|x\right|\le ⟨value⟩$.

$\left|x\right|$ is too large, the function returns the amplitude of the $J_0$ oscillation, $\sqrt{2/\left(\pi \left|x\right|\right)}$.

Notes

bessel_j0_real evaluates an approximation to the Bessel function of the first kind $J_0\left(x\right)$.

Note: $J_0(-x)=J_0\left(x\right)$, so the approximation need only consider $x\ge 0$.

The function is based on three Chebyshev expansions:

For $0<x\le 8$,

$$J_0\left(x\right)={\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)\text{, with}t=2{\left(\frac{x}{8}\right)}^2-1\text{.}$$

For $x>8$,

$$J_0\left(x\right)=\sqrt{\frac{2}{\pi x}}\{P_0\left(x\right)\cos (x-\frac{\pi }{4})-Q_0\left(x\right)\sin (x-\frac{\pi }{4})\}\text{,}$$

where $P_0\left(x\right)={{\sum }^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)$,

and $Q_0\left(x\right)=\frac{8}{x}{{\sum }^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)$,

with $t=2{\left(\frac{8}{x}\right)}^2-1$.

For $x$ near zero, $J_0\left(x\right)\simeq 1$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.

For very large $x$, it becomes impossible to provide results with any reasonable accuracy (see Accuracy), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of $J_0\left(x\right)$; only the amplitude, $\sqrt{\frac{2}{\pi \left|x\right|}}$, can be determined and this is returned on failure. The range for which this occurs is roughly related to machine precision; the function will fail if $\left|x\right|\gtrsim 1/\text{machine precision}$.

References

NIST Digital Library of Mathematical Functions

Clenshaw, C W, 1962, Chebyshev Series for Mathematical Functions, Mathematical tables, HMSO