import numpy as np
import matplotlib.pyplot as plt
from scipy.special import gammaincc, gamma

def rho(r, N):
    # Upper incomplete gamma Γ(N, N r²)
    upper_inc_gamma = gammaincc(N, N * r**2) * gamma(N)
    return upper_inc_gamma / (np.pi * gamma(N))

r = np.linspace(0, 2, 400)
Ns = [1, 5, 20, 100]

plt.figure(figsize=(6, 4))

for N in Ns:
    plt.plot(r, np.pi * rho(r, N), label=f"{N}")

plt.xlabel("|z|")
ho(z)$")
plt.title("Eigenvalue density of $\frac{1}{\sqrt{N}} G_N$")
plt.xlim(0, 1.5)
plt.ylim(0, 1.1)
plt.legend()
plt.grid(True)
plt.savefig("eigenvalue density of Ginibre.svg")
plt.show()