Appendix
Transform Tables

Table of Representative Unilateral Laplace Transform Pairs

\(x(t)\)	\(X(s)\)
\(\delta(t)\)	\(1\)
\(\delta(t - t_0)\)	\(e^{-s\, t_0}\) for \(t_0 > 0\)
\(u(t)\)	\(\frac{1}{s}\) for \(\text{Re}\{s\} > 0\)
\(e^{-at}u(t)\)	\(\frac{1}{s+a}\) for \(\text{Re}\{s\} > -a\)
\(te^{-at}u(t)\)	\(\frac{1}{(s + a)^2}\) for \(\text{Re}\{s\} > -a\)
\(e^{s_0 t}u(t)\)	\(\frac{1}{s+s_0}\) for \(\text{Re}\{s\} > \text{Re}\{s_0\}\)
\(\cos(\omega_0 t)u(t)\)	\(\frac{s}{s^2 + \omega_0^2}\) for \(\text{Re}\{s\} > 0\)
\(\sin(\omega_0 t)u(t)\)	\(\frac{\omega_0}{s^2 + \omega_0^2}\) for \(\text{Re}\{s\} > 0\)
\(e^{-at}\cos(\omega_0 t)u(t)\)	\(\frac{s + a}{(s + a)^2 + \omega_0^2}\) for \(\text{Re}\{s\} > -a\)
\(e^{-at}\sin(\omega_0 t)u(t)\)	\(\frac{\omega_0}{(s+a)^2 + \omega_0^2}\) for \(\text{Re}\{s\} > -a\)
\(re^{-at}\cos(\omega_0 t + \theta)u(t)\)	\(\frac{(r\cos(\theta))s + \left(ar\cos(\theta) - \omega_0 r\sin(\theta)\right)}{s^2 + 2as + \left(a^2 + \omega_0^2\right)}\) for \(\text{Re}\{s\} > -a\)
\(e^{-at}\left[A\cos(\omega_0 t) + \frac{B-Aa}{\omega_0}\sin(\omega_0 t)\right]u(t)\)	\(\frac{As + B}{s^2 + 2as + c}\) for \(\text{Re}\{s\} > -a\)
\(\omega_0 = \sqrt{c-a^2}\)

Table of Representative Unilateral Z Transform Pairs

\(x[n]\)	\(X(z)\)
\(\delta[n]\)	\(1\)
\(\delta[n - m]\)	\(z^{-m}\) for \(m > 0\)
\(u[n]\)	\(\frac{z}{z-1}\) for \(|z| > 1\)
\(n\, u[n]\)	\(\frac{z}{(z-1)^2}\) for \(|z| > 1\)
\(\left(a\right)^n\, u[n]\)	\(\frac{z}{z-a}\) for \(|z| > a\)
\(\left(a\right)^{n-1}\, u[n-1]\)	\(\frac{1}{z-a}\) for \(|z| > a\)
\(n\left(a\right)^n\, u[n]\)	\(\frac{az}{(z-a)^2}\) for \(|z| > a\)
\(|a|^n\, \cos(\omega_0\, n)\, u[n]\)	\(\frac{z\left(z-|a|\cos(\omega_0)\right)}{z^2 - \left(2|a|\cos(\omega_0)\right)\, z + |a|^2}\)
\(|a|^n\, \sin(\omega_0\, n)\, u[n]\)	\(\frac{z|a|\sin(\omega_0)}{z^2 - \left(2|a|\cos(\omega_0)\right)\, z + |a|^2}\)
\(r\left|b\right|^n\, \cos(\omega_0 n + \theta)\, u[n]\)	\(\frac{z(Az+B)}{z^2 + 2az+|b|^2}\)
\(r = \sqrt{\frac{A^2|b|^2 + B^2 -2AaB}{|b|^2 - a^2}}\)
\(\omega_0 = \cos^{-1}\left( \frac{-a}{|b|}\right)\)
\(\theta = \tan^{-1}\left( \frac{Aa-B}{A\sqrt{|b|^2 -a^2}}\right)\)