I have a function that I want to find its roots. I could write a program to figure out its roots but the point is, each time that I want to find the other root I should give it an initial value manually which I do not want to do that. I want to have all the roots in a list since I want to do some operations on the roots after finding them. This is my code:

import math
import scipy
import scipy.optimize
c = 5
alambda = 1
rho = 0.8
b = rho * c / alambda
def f(zeta):y = ((zeta**c)*(math.exp((alambda*b)*(1-zeta)))) - 1return yprint scipy.optimize.newton(f,  -1)

For integer c, the roots of the function are given by the Lambert W function, https://en.wikipedia.org/wiki/Lambert_W_function

from numpy import exp, pi
from scipy.special import lambertw
c = 3
alambda = 1.234
rho = 0.8
b = rho * c / alambda
def f(zeta):y = ((zeta**c)*(exp((alambda*b)*(1-zeta)))) - 1return y
def zeta_root(k, n):a = alambdareturn -c/(a*b) * lambertw(-a*b/c * exp(-(a*b+2j*pi*n)/c), k=k)
for k in range(-20, 20):# also n can be any integer; probably reproduces the same root set# as varying kzeta = zeta_root(k, 3)print("k={0}, zeta={1}, error={2}".format(k, zeta, abs(f(zeta))))

The equation has an infinite number of complex-valued roots. The k=0 and k=-1 roots which correspond to real branches of the Lambert W function may be real-valued.

For non-integer c the situation seems a bit more complicated due to the extra branch cut, although at least the real-valued positive roots should be captured.