Error Function, and variants

Computes the error function, or its inverse, based on the normal distribution. Also computes the complement of the error function, or its inverse,

erf(x, inverse = FALSE)
erfc(x, inverse = FALSE)

Arguments

x: Numeric.
inverse: Logical. Of length 1.

Details

$Erf(x)$ is defined as $$Erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-t^2) dt$$ so that it is closely related to pnorm. The inverse function is defined for $x$ in $(-1,1)$.

Value

Returns the value of the function evaluated at x.

References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications Inc.

Author

T. W. Yee

Note

Some authors omit the term $2/\sqrt{\pi}$ from the definition of $Erf(x)$. Although defined for complex arguments, this function only works for real arguments.

The complementary error function $erfc(x)$ is defined as $1-erf(x)$, and is implemented by erfc. Its inverse function is defined for $x$ in $(0,2)$.

Examples

if (FALSE) { # \dontrun{
curve(erf,   -3, 3, col = "orange", ylab = "", las = 1)
curve(pnorm, -3, 3, add = TRUE, col = "blue", lty = "dotted", lwd = 2)
abline(v = 0, h = 0, lty = "dashed")
legend("topleft", c("erf(x)", "pnorm(x)"), col = c("orange", "blue"),
       lty = c("solid", "dotted"), lwd = 1:2) } # }