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====== List of Integral / Space Transforms ======

===== Laplace Transforms =====

$$F(s) = \int_{-\infty}^{\infty}\,f(x)e^{st}\,dt$$

^ $f(t)$                         ^ $F(s)$                          ^
| $$1$$                          | $$\frac{1}{s}$$                 |
| $$t^n\,,(n=0,1,2,...)$$        | $$\frac{n!}{s^{n+1}}$$          |
| $$t^nf(t)$$                    | $$(-1)\stackrel{n}{F}(s)$$      |
| $$e^{at}$$                     | $$\frac{1}{s-a}$$               |
| $$e^{at}f(t)$$                 | $$F(s-a)$$                      |
| $$te^{at}$$                    | $$\frac{1}{(s-a)^2}$$           |
| $$t^ne^{at}$$                  | $$\frac{n!}{(s-a)^{n+1}}$$      |
| $$\frac{e^{at}-e^{bt}}{a-b}$$  | $$\frac{1}{(s-a)(s-b)}$$        |
| $$\frac{ae^{at}-e^{bt}}{a-b}$$ | $$\frac{s}{(s-a)(s-b)}$$        |
| $$\sin(kt)$$                   | $$\frac{k}{s^2+k^2}$$           |
| $$\cos(kt)$$                   | $$\frac{s}{s^2+k^2}$$           |
| $$t\sin(kt)$$                  | $$\frac{2ks}{(s^2+k^2)^2}$$     |
| $$t\cos(kt)$$                  | $$\frac{s^2-k^2}{(s^2+k^2)^2}$$ |
| $$e^{at}\sin(kt)$$             | $$\frac{k}{(s-a)^2-k^2}$$       |
| $$e^{at}\cos(kt)$$             | $$\frac{s-a}{(s-a)^2+k^2}$$     |
| $$\frac{\sin{at}}{t}$$         | $$\text{atan}\left(\frac{a}{s}\right)$$    |
| $$\dot{f}(t)$$                 | $$sF(s) -f(0)$$                 |
| $$\int_0^t{\,f(t)\,dt}$$       | $$\frac{1}{s}\,F(s)$$           |
| $$f(t) * g(t)$$                | $$F(s)G(s)$$                    |
| $$\delta(t)$$                  | $$1$$                           |
| $$\delta(t-t_0)$$              | $$e^{-st_0}$$                   |
| $$u(t-a)$$                     | $$\frac{e^{-as}}{s}$$           |
| $$u(t-a)f(t-a)$$               | $$e^{a-s}F(s)$$                 |

===== Z Transforms =====

$$F[z] = \sum_{n=-\infty}^{\infty}{f[n]z^{-n}}$$


  * Geometric Series: 
    $$\sum_{n=0}^{\infty}{a^n} = \frac{1}{1-a}$$

For most transforms, taking the inversion of the coefficients and arguments leads to the same transform, but the ROC is inverted. e.g.

$$\mathcal{Z}\{u[n]\}=\frac{z}{z-1}\quad ROC\,|z|>1$$
$$\mathcal{Z}\{-u[-n-1]\}=\frac{z}{z-1}\quad ROC\,|z|<1$$


^ $$f[n]$$                    ^ $$F[z]$$                                                                                                                 ^ $$ROC$$               ^            
| $$x[n]$$                    | $$X[z]$$                                                                                                                 | $$r_2 < |z| < r_1$$   |
| $$x[n-k]$$                  | $$z^{-k}X[z]$$                                                                                                           | $$|z|\neq 0$$         |
| $$x[n+k]$$                  | $$z^{k}X[z]$$                                                                                                            | $$|z|\neq \infty$$    |
| $$a^nx[n]$$                 | $$X\left[\frac{z}{a}\right]$$                                                                                            | $$|a|r_2<|z|<|a|r_1$$ |
| $$\delta [n]$$              | $$1$$                                                                                                                    | $$\text{All } z$$     |
| $$u[n]$$                    | $$\frac{z}{z-1}=\frac{1}{1-z^{-1}}$$                                                                                     | $$|z| > 1$$           |
| $$a^nu[n]$$                 | $$\frac{z}{z-a}=\frac{1}{1-az^{-1}}$$                                                                                    | $$|z| > |a|$$         |
| $$nu[n]$$(($ramp(t)$))      | $$\frac{z}{(z-1)^2}=\frac{z^{-1}}{(1-z^{-1})^2}$$                                                                        | $$|z| > 1$$           |
| $$n^2u[n]$$                 | $$\frac{z(z+1)}{(z-1)^3}=\frac{1+z^{-1}}{z(1-z^{-1})^2}$$                                                                | $$|z| > 1$$           |
| $$na^nu[n]$$                | $$\frac{za}{(z-a)^2}=\frac{az^{-1}}{(1-az^{-1})^2}$$                                                                     | $$|z| > |a|$$         |
| $$\cos(\omega_0 n)u[n]$$    | $$\frac{z(z-\cos(\omega_0))}{z^2-2z\cos(\omega_0)+1}=\frac{1-z^1\cos(\omega_0)}{1-2z^{-1}\cos(\omega_0)+z^{-2}}$$        | $$|z| > 1$$           |
| $$\sin(\omega_0 n)u[n]$$    | $$\frac{z\sin(\omega_0)}{z^2-2z\cos(\omega_0)+1}=\frac{z^{-1}\sin(\omega_0)}{1-2z^{-1}\cos(\omega_0)+z^{-2}}$$           | $$|z| > 1$$           |
| $$a^n\cos(\omega_0 n)u[n]$$ | $$\frac{z(z-a\cos(\omega_0))}{a^2-2az\cos(\omega_0)+1}=\frac{1-az^1\cos(\omega_0)}{1-2az^{-1}\cos(\omega_0)+a^2z^{-2}}$$ | $$|z| > 1$$           |
| $$a^n\sin(\omega_0 n)u[n]$$ | $$\frac{az\sin(\omega_0)}{a^2-2az\cos(\omega_0)+z^2}=\frac{az^{-1}\sin(\omega_0)}{1-2az^{-1}\cos(\omega_0)+a^2z^{-2}}$$  | $$|z| > 1$$           |
| $$e^{at}$$                  | $$\frac{z}{z-e^{at}}$$                                                                                                   | $$|z| > 1$$           |
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