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
New 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)
satisfiesw + 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)