naginterfaces.library.specfun.bessel_​y0_​real_​vector

naginterfaces.library.specfun.bessel_y0_real_vector(x)

bessel_y0_real_vector returns an array of values of the Bessel function $Y_0\left(x\right)$.

For full information please refer to the NAG Library document for s17aq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s17aqf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The argument $x_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

Returns

ffloat, ndarray, shape $\left(n\right)$

$Y_0\left(x_i\right)$, the function values.

ivalidint, ndarray, shape $\left(n\right)$

$ivalid[\text{i}-1]$ contains the error code for $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, $x_i$ is too large. $f[\text{i}-1]$ contains the amplitude of the $Y_0$ oscillation, $\sqrt{\frac{2}{\pi x_i}}$.

$ivalid[i-1]=2$

On entry, $x_i\le 0.0$, $Y_0$ is undefined. $f[\text{i}-1]$ contains $0.0$.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $x$ was invalid.

Check $ivalid$ for more information.

Notes

bessel_y0_real_vector evaluates an approximation to the Bessel function of the second kind $Y_0\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Note: $Y_0\left(x\right)$ is undefined for $x\le 0$ and the function will fail for such arguments.

The function is based on four Chebyshev expansions:

For $0<x\le 8$,

$$Y_0\left(x\right)=\frac{2}{\pi }ln\left(x\right){\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)+{\sum ^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)\text{, with}t=2{\left(\frac{x}{8}\right)}^2-1\text{.}$$

For $x>8$,

$$Y_0\left(x\right)=\sqrt{\frac{2}{\pi x}}\{P_0\left(x\right)\sin (x-\frac{\pi }{4})+Q_0\left(x\right)\cos (x-\frac{\pi }{4})\}$$

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

and $Q_0\left(x\right)=\frac{8}{x}{{\sum }^{\prime }}_{r=0}{d}_r{T}_r\left(t\right),\text{with}t=2{\left(\frac{8}{x}\right)}^2-1\text{.}$

For $x$ near zero, $Y_0\left(x\right)\simeq \frac{2}{\pi }(ln\left(\frac{x}{2}\right)+\gamma )$, where $\gamma$ denotes Euler’s constant. 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 $Y_0\left(x\right)$; only the amplitude, $\sqrt{\frac{2}{\pi n}}$, 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 $x\gtrsim 1/\text{machine precision}$.

References

NIST Digital Library of Mathematical Functions

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