Invert the following to obtain pnt Pz t z1 z2 e2 lambda tN

Invert the following to obtain p_n(t) P(z, t) = [z/1 - z^2 e^-2 lambda t]^N.

Solution

In Z transform domain, this is in the form of sum of an infinite GP which is the standard form. With first term, a = Z and the common ratio, r= Z²e^-2.lambda.t

Thus the series is= Z + Z³e^-2.lambda.t + Z⁵e^-4.lambda.t.... So on upto infinity.

= Z (1 + Z²e^-2.lambda.t .....).

So, p_n = derivative of ( e^{-2.n.t.lambda}) = -2.t.lambda.e^{-2.n.t.lambda}