Characteristic Function

from sympy import symbols, exp, diff, simplify, I
w = symbols('w')

def raw_moment(cf, k):
    return simplify((-I)**k * diff(cf, w, k).subs(dict(w=0)))

def central_moment(cf, k):
    m = raw_moment(cf, 1)
    return simplify(raw_moment(cf/exp(I*w*m),k))

n1, t1 = symbols('n1, t1')
cf1 = exp(n1*(1/(1 - I*w*t1) - 1))
cf1

raw_moment(cf1, 1)

central_moment(cf1, 2)

central_moment(cf1, 3)

n1, t1, n2, t2, t0, a, b = symbols('n1, t1, n2, t2, t0, a, b')
cf2 = exp(a*n1*(1/(1 - I*w*t1) - 1) + b*n2*(1/(1 - I*w*t2) - 1) + I*w*t0)
cf2

raw_moment(cf2, 1)

central_moment(cf2, 2)