Error Functions and Inverses (Matlab Style)

The error or Phi function is a variant of the cumulative normal (or Gaussian) distribution.

erf(x)
erfinv(y)
erfc(x)
erfcinv(y)
erfcx(x)

erfz(z)
erfi(z)

Arguments

x, y

vector of real numbers.

z

real or complex number; must be a scalar.

Details

erf and erfinv are the error and inverse error functions.
erfc and erfcinv are the complementary error function and its inverse.
erfcx is the scaled complementary error function.
erfz is the complex, erfi the imaginary error function.

Value

Real or complex number(s), the value(s) of the function.

Author

First version by Hans W Borchers; vectorized version of erfz by Michael Lachmann.

Note

For the complex error function we used Fortran code from the book S. Zhang & J. Jin “Computation of Special Functions” (Wiley, 1996).

See also

pnorm

Examples

  x <- 1.0
  erf(x); 2*pnorm(sqrt(2)*x) - 1
#> [1] 0.8427008
#> [1] 0.8427008
# [1] 0.842700792949715
# [1] 0.842700792949715
  erfc(x); 1 - erf(x); 2*pnorm(-sqrt(2)*x)
#> [1] 0.1572992
#> [1] 0.1572992
#> [1] 0.1572992
# [1] 0.157299207050285
# [1] 0.157299207050285
# [1] 0.157299207050285
  erfz(x)
#> [1] 0.8427008
# [1] 0.842700792949715
  erfi(x)
#> [1] 1.650426
# [1] 1.650425758797543