In conventional mathematical notation, your equation is
The SciPy fsolve function searches for a point at which a given expression equals zero (a “zero” or “root” of the expression). You’ll need to provide fsolve with an initial guess that’s “near” your desired solution. A good way to find such an initial guess is to just plot the expression and look for the zero crossing.
#!/usr/bin/python
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
# Define the expression whose roots we want to find
a = 0.5
R = 1.6
func = lambda tau : R - ((1.0 - np.exp(-tau))/(1.0 - np.exp(-a*tau)))
# Plot it
tau = np.linspace(-0.5, 1.5, 201)
plt.plot(tau, func(tau))
plt.xlabel("tau")
plt.ylabel("expression value")
plt.grid()
plt.show()
# Use the numerical solver to find the roots
tau_initial_guess = 0.5
tau_solution = fsolve(func, tau_initial_guess)
print "The solution is tau = %f" % tau_solution
print "at which the value of the expression is %f" % func(tau_solution)