import escape as esc
esc.require('0.9.1')
from escape.utils import show
import numpy as np

Gamma Functors

True gamma

The true gamma is defined as $\Gamma(z)=\int_0^\infty t^{z-1}\exp(-t)dt$ When calculated out of allowed domains the NaN values are returned.

X=esc.var("X")
tg = esc.tgamma(X)
coords=np.linspace(-6, 6, 1000)
show(tg, coordinates=coords, xlabel="X", ylabel="Y", title="True Gamma")

Incomplete gamma functions

There are four incomplete gamma functions: two are known as regularized incomplete gamma functions, i.e. normalized, that return values in the range $[0, 1]$, and other two are non-normalised and return values in the range $[0, \Gamma(a)]$.

The first type is defined as following:

$P(z, a)=\frac{1}{\Gamma(a)}\int_0^z t^{a-1}\exp(-t)dt$

a = esc.par("a", 1)
gp = esc.gamma_p(a, X, normalized=True)
#if you don't need normalization, set the *normaliazed* argument to False.
coords=np.linspace(0, 6, 1000)
show(gp, coordinates=coords, xlabel="X", ylabel="Y", title="P(X, a)")

The second type is defined as following:

$Q(z, a)=\frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}\exp(-t)dt$

gq = esc.gamma_q(a, X, normalized=True)
#if you don't need normalization, set the *normaliazed* argument to False.
coords=np.linspace(0, 6, 1000)
show(gq, coordinates=coords, xlabel="X", ylabel="Y", title="Q(X, a)")