naginterfaces.library.specfun.bessel_​y1_​real

naginterfaces.library.specfun.bessel_y1_real(x)

bessel_y1_real returns the value of the Bessel function $Y_1\left(x\right)$.

For full information please refer to the NAG Library document for s17ad

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s17adf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

y1float

The value of the Bessel function $Y_1\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\le ⟨value⟩$.

$x$ is too large, the function returns the amplitude of the $Y_1$ oscillation, $\sqrt{2/\left(\pi x\right)}$.

(errno $2$)

On entry, $x=⟨value⟩$.

Constraint: $x>0.0$.

$Y_1$ is undefined, the function returns zero.

(errno $3$)

$x$ is too close to zero and there is danger of overflow, $x=⟨value⟩$.

Constraint: $x>⟨value⟩$.

The function returns the value of $Y_1\left(x\right)$ at the smallest valid argument.

Notes

bessel_y1_real evaluates an approximation to the Bessel function of the second kind $Y_1\left(x\right)$.

Note: $Y_1\left(x\right)$ is undefined for $x\le 0$ and the function will fail for such arguments.

The function is based on four Chebyshev expansions:

For $0<x\le 8$,

$$Y_1\left(x\right)=\frac{2}{\pi }ln\left(x\right)\frac{x}{8}{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)-\frac{2}{\pi x}+\frac{x}{8}{\sum ^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)\text{, with}t=2{\left(\frac{x}{8}\right)}^2-1\text{.}$$

For $x>8$,

$$Y_1\left(x\right)=\sqrt{\frac{2}{\pi x}}\{P_1\left(x\right)\sin (x-3\frac{\pi }{4})+Q_1\left(x\right)\cos (x-3\frac{\pi }{4})\}$$

where $P_1\left(x\right)={{\sum }^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)$,

and $Q_1\left(x\right)=\frac{8}{x}{{\sum }^{\prime }}_{r=0}{d}_r{T}_r\left(t\right)$, with $t=2{\left(\frac{8}{x}\right)}^2-1$.

For $x$ near zero, $Y_1\left(x\right)\simeq -\frac{2}{\pi x}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For extremely small $x$, there is a danger of overflow in calculating $-\frac{2}{\pi x}$ and for such arguments the function will fail.

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 $Y_1\left(x\right)$; only the amplitude, $\sqrt{\frac{2}{\pi x}}$, 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