scipy.special.wrightomega#

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

Wright Omega function.

Defined as the solution to

\[\omega + \log(\omega) = z\]

where \(\log\) is the principal branch of the complex logarithm.

Parameters:

zarray_like: Points at which to evaluate the Wright Omega function
outndarray, optional: Optional output array for the function values

Returns:

omegascalar or ndarray: Values of the Wright Omega function

See also

lambertw: The Lambert W function

Notes

Added in version 0.19.0.

The function can also be defined as

\[\omega(z) = W_{K(z)}(e^z)\]

where \(K(z) = \lceil (\Im(z) - \pi)/(2\pi) \rceil\) is the unwinding number and \(W\) is the Lambert W function.

The implementation here is taken from [1].

References

[1]

Lawrence, Corless, and Jeffrey, “Algorithm 917: Complex Double-Precision Evaluation of the Wright \(\omega\) Function.” ACM Transactions on Mathematical Software, 2012. DOI:10.1145/2168773.2168779.

Examples

>>> import numpy as np
>>> from scipy.special import wrightomega, lambertw

>>> wrightomega([-2, -1, 0, 1, 2])
array([0.12002824, 0.27846454, 0.56714329, 1.        , 1.5571456 ])

Complex input:

>>> wrightomega(3 + 5j)
(1.5804428632097158+3.8213626783287937j)

Verify that wrightomega(z) satisfies w + log(w) = z:

>>> w = -5 + 4j
>>> wrightomega(w + np.log(w))
(-5+4j)

Verify the connection to lambertw:

>>> z = 0.5 + 3j
>>> wrightomega(z)
(0.0966015889280649+1.4937828458191993j)
>>> lambertw(np.exp(z))
(0.09660158892806493+1.4937828458191993j)

>>> z = 0.5 + 4j
>>> wrightomega(z)
(-0.3362123489037213+2.282986001579032j)
>>> lambertw(np.exp(z), k=1)
(-0.33621234890372115+2.282986001579032j)