naginterfaces.library.specfun.bessel_​y0_​real

naginterfaces.library.specfun.bessel_y0_real(x)

bessel_y0_real returns the value of the Bessel function $Y_0\left(x\right)$.

For full information please refer to the NAG Library document for s17ac

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s17acf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

y0float

The value of the Bessel function $Y_0\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\le ⟨value⟩$.

(errno $2$)

On entry, $x=⟨value⟩$.

Constraint: $x>0.0$.

Notes

bessel_y0_real evaluates an approximation to the Bessel function of the second kind $Y_0\left(x\right)$.

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 $0<x\le 8$,

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

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