Skip to main content
`Ctrl`+`K`

# scipy.special.erf

scipy.special.erf(z, out=None) = <ufunc 'erf'>

  Returns the error function of complex argument.

  It is defined as `2/sqrt(pi)*integral(exp(-t**2), t=0..z)`.

  Parameters:
    xndarray

      Input array.

    outndarray, optional

      Optional output array for the function values

  Returns:
    resscalar or ndarray

      The values of the error function at the given points x.

  See also

  `erfc`, `erfinv`, `erfcinv`, `wofz`, `erfcx`, `erfi`

  Notes

  The cumulative of the unit normal distribution is given by `Phi(z) = 1/2[1 + erf(z/sqrt(2))]`.

  `erf` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ✅ | n/a

  See Support for the array API standard for more information.

  References

  [1]

  https://en.wikipedia.org/wiki/Error_function

  [2]

  Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. http://www.math.sfu.ca/~cbm/aands/page_297.htm

  [3]

  Steven G. Johnson, Faddeeva W function implementation. http://ab-initio.mit.edu/Faddeeva

  Examples

      >>> import numpy as np
      >>> from scipy import special
      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(-3, 3)
      >>> plt.plot(x, special.erf(x))
      >>> plt.xlabel('$x$')
      >>> plt.ylabel('$erf(x)$')
      >>> plt.show()