naginterfaces.library.specfun.airy_​ai_​complex

naginterfaces.library.specfun.airy_ai_complex(deriv, z, scal)

airy_ai_complex returns the value of the Airy function $Ai\left(z\right)$ or its derivative ${\text{Ai}}^{\prime }\left(z\right)$ for complex $z$, with an option for exponential scaling.

For full information please refer to the NAG Library document for s17dg

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s17dgf.html

Parameters

derivstr, length 1

Specifies whether the function or its derivative is required.

$deriv=\text{'F'}$

$Ai\left(z\right)$ is returned.

$deriv=\text{'D'}$

${\text{Ai}}^{\prime }\left(z\right)$ is returned.

zcomplex

The argument $z$ of the function.

scalstr, length 1

The scaling option.

$scal=\text{'U'}$

The result is returned unscaled.

$scal=\text{'S'}$

The result is returned scaled by the factor $e^{2z\sqrt{z}/3}$.

Returns

aicomplex

The required function or derivative value.

nzint

Indicates whether or not $ai$ is set to zero due to underflow. This can only occur when $scal=\text{'U'}$.

$nz=0$

$ai$ is not set to zero.

$nz=1$

$ai$ is set to zero.

Raises

NagValueError

(errno $1$)

On entry, $deriv$ has an illegal value: $deriv=⟨value⟩$.

(errno $1$)

On entry, $scal$ has an illegal value: $scal=⟨value⟩$.

(errno $2$)

No computation because $Re\left(\omega \right)$ too large, where $\omega =\left(2/3\right)\times z^{\left(3/2\right)}$.

(errno $4$)

No computation because $\left|z\right|=⟨value⟩>⟨value⟩$.

(errno $5$)

No computation – algorithm termination condition not met.

Warns

NagAlgorithmicWarning

(errno $3$)

Results lack precision because $\left|z\right|=⟨value⟩>⟨value⟩$.

Notes

airy_ai_complex returns a value for the Airy function $Ai\left(z\right)$ or its derivative ${\text{Ai}}^{\prime }\left(z\right)$, where $z$ is complex, $-\pi <argz\le \pi$. Optionally, the value is scaled by the factor $e^{2z\sqrt{z}/3}$.

The function is derived from the function CAIRY in Amos (1986). It is based on the relations $Ai\left(z\right)=\frac{\sqrt{z}{K}_{1/3}\left(w\right)}{\pi \sqrt{3}}$, and ${\text{Ai}}^{\prime }\left(z\right)=\frac{-z{K}_{2/3}\left(w\right)}{\pi \sqrt{3}}$, where $K_{\nu }$ is the modified Bessel function and $w=2z\sqrt{z}/3$.

For very large $\left|z\right|$, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller $\left|z\right|$, the computation is performed but results are accurate to less than half of machine precision. If $Re\left(w\right)$ is too large, and the unscaled function is required, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the function.

References

NIST Digital Library of Mathematical Functions

Amos, D E, 1986, Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order, ACM Trans. Math. Software (12), 265–273