freealg.distributions.Wigner

class freealg.distributions.Wigner(r)

Wigner semicircle distribution.

Notes

The Wigner distribution has the absolutely-continuous density

\[\mathrm{d} \rho(x) = \frac{1}{2 \pi} \sqrt{(4 - x^2} \mathbf{1}_{x \in [\lambda_{-}, \lambda_{+}]} \mathrm{d}{x}\]

with \(\lambda_{\pm} = \pm 2\) being the edges of the support

References

[1]

Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics, 62(3), 548-564.421

Examples

>>> from freealg.distributions import Wigner
>>> wg = Wigner(1)

Methods

density([x, plot, latex, save, eig])	Density of distribution.
hilbert([x, plot, latex, save])	Hilbert transform of the distribution.
stieltjes([x, y, plot, on_disk, latex, save])	Stieltjes transform of distribution.
sample(size[, x_min, x_max, method, seed, ...])	Sample from distribution.
matrix(size[, seed])	Generate matrix with the spectral density of the distribution.