Question

If e^(rt) is given by e^[(A+iu)t] = e^(At)cos(ut)+ie^(At)sin(ut), show that e^[(r1+r2)t] = e^(r1t)e^(r2t) for any complex numbers r1 and r2

Answer 1

e^[(A1+iu1)t] = e^(A1t)cos(u1t)+ie^(A1t)sin(u1t)
e^[(A2+iu2)t] = e^(A2t)cos(u2t)+ie^(A2t)sin(u2t)
multiply them
then simplify