In Chapter 7., we found that the step response of a first-order circuit can be written as

where \(f(0)\) is the initial value, \(f(\infty)\) is the final value and \(T_{\mathrm{C}}\) is the time constant. Show that the corresponding transform has the form

and relate the time-domain parameters \(f(0), f(\infty)\), and \(T_{\mathrm{C}}\) to the \(s\)-domain parameters \(K, \gamma\), and \(\alpha\).

f(t) = f(00)+[f(0) - f(oo)] e-/Tc