from numpy import arange, sin, pi
from scipy.signal import hilbert
import matplotlib.pyplot as plt
plt.rc('svg', fonttype='none')

dt = 1e-3
t = arange(dt, 3, dt)
y1 = -(1.2 / t) - t + 3
y1[y1 < 0] = 0
y2 = sin(2 * 3 * pi * t)
y3 = y1 * y2
y4 = abs(hilbert(y3))

fig, ax = plt.subplots(1,1,figsize=(8,5))
ax.plot(t, y3, color='blue')
ax.plot(t, y4, color='red')
ax.grid(True)
ax.minorticks_on()
ax.set_xlim([0,3])
ax.set_xlabel(r'$t$')
ax.set_ylabel(r'$y$')
fig.savefig('analytic.svg', bbox_inches='tight')