scipy.special.erf#

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

Error function of real or complex argument.

\[\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt\]

Parameters:

zndarray: 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 z.

See also

erfc, erfcx, erfi, erfinv, erfcinv, wofz

Notes

The cumulative distribution function (CDF) of the standard normal distribution can be expressed in terms of the error function as

\[\Phi(z) = \frac{1}{2} \left[1 + \operatorname{erf} \left(\frac{z}{\sqrt{2}}\right)\right]\]

Array API Standard Support

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.

[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
>>> z = np.linspace(-3, 3)
>>> plt.plot(z, special.erf(z))
>>> plt.xlabel('$z$')
>>> plt.ylabel('$erf(z)$')
>>> plt.show()