s15af returns a value for Dawson's Integral, $F\left(x\right)$.

# Syntax

C#
```public static double s15af(
double x
)```
Visual Basic
```Public Shared Function s15af ( _
x As Double _
) As Double```
Visual C++
```public:
static double s15af(
double x
)```
F#
```static member s15af :
x : float -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.

#### Return Value

s15af returns a value for Dawson's Integral, $F\left(x\right)$.

# Description

s15af evaluates an approximation for Dawson's Integral
 $Fx=e-x2∫0xet2dt.$
The method is based on two Chebyshev expansions:
For $0<\left|x\right|\le 4$,
 $Fx=x∑r=0′arTrt, where t=2x42-1.$
For $\left|x\right|>4$,
 $Fx=1x∑r=0′brTrt, where t=24x2-1.$
For $\left|x\right|$ near zero, $F\left(x\right)\simeq x$, and for $\left|x\right|$ large, $F\left(x\right)\simeq \frac{1}{2x}$. These approximations are used for those values of $x$ for which the result is correct to machine precision.

# References

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

None.

# Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is considerably greater than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃x1-2xFxFxδ.$
The following graph shows the behaviour of the error amplification factor $\left|\frac{x\left(1-2xF\left(x\right)\right)}{F\left(x\right)}\right|$:
Figure 1
However if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ somewhat larger than the above relation indicates. In fact $\epsilon$ will be largely independent of $x$ or $\delta$, but will be of the order of a few times the machine precision.

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#): s15afe.cs

Example program data: s15afe.d

Example program results: s15afe.r