﻿ s17aj Method
s17aj returns a value of the derivative of the Airy function $\mathrm{Ai}\left(x\right)$.

# Syntax

C#
```public static double s17aj(
double x,
out int ifail
)```
Visual Basic
```Public Shared Function s17aj ( _
x As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s17aj(
double x,
[OutAttribute] int% ifail
)```
F#
```static member s17aj :
x : float *
ifail : int byref -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s17aj returns a value of the derivative of the Airy function $\mathrm{Ai}\left(x\right)$.

# Description

s17aj evaluates an approximation to the derivative of the Airy function $\mathrm{Ai}\left(x\right)$. It is based on a number of Chebyshev expansions.
For $x<-5$,
 $Ai′x=-x4atcos z+btζsin z,$
where $z=\frac{\pi }{4}+\zeta$, $\zeta =\frac{2}{3}\sqrt{-{x}^{3}}$ and $a\left(t\right)$ and $b\left(t\right)$ are expansions in variable $t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For $-5\le x\le 0$,
 $Ai′x=x2ft-gt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Ai′x=e-11x/8yt,$
where $y\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-1$.
For $4.5\le x<9$,
 $Ai′x=e-5x/2vt,$
where $v\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-3$.
For $x\ge 9$,
 $Ai′x=x4e-zut,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $u\left(t\right)$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.
For $\left|x\right|<\text{}$ the square of the machine precision, the result is set directly to ${\mathrm{Ai}}^{\prime }\left(0\right)$. This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the method must fail. This occurs for $x<-{\left(\frac{\sqrt{\pi }}{\epsilon }\right)}^{4/7}$, where $\epsilon$ is the machine precision.
For large positive arguments, where ${\mathrm{Ai}}^{\prime }$ decays in an essentially exponential manner, there is a danger of underflow so the method must fail.

# References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
x is too large and positive. On failure, the method returns zero. (see the Users' Note for your implementation for details)
${\mathbf{ifail}}=2$
x is too large and negative. On failure, the method returns zero. See also the Users' Note for your implementation.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential in character and here relative error is needed. The absolute error, $E$, and the relative error, $\epsilon$, are related in principle to the relative error in the argument, $\delta$, by
 $E≃x2Aixδ ε≃x2AixAi′xδ.$
In practice, approximate equality is the best that can be expected. When $\delta$, $\epsilon$ or $E$ is of the order of the machine precision, the errors in the result will be somewhat larger.
For small $x$, positive or negative, errors are strongly attenuated by the function and hence will be roughly bounded by the machine precision.
For moderate to large negative $x$, the error, like the function, is oscillatory; however the amplitude of the error grows like
 $x7/4π.$
Therefore it becomes impossible to calculate the function with any accuracy if ${\left|x\right|}^{7/4}>\frac{\sqrt{\pi }}{\delta }$.
For large positive $x$, the relative error amplification is considerable:
 $εδ≃x3.$
However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification is limited.

None.

None.

# Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

Example program (C#): s17aje.cs

Example program data: s17aje.d

Example program results: s17aje.r