Table of Transform Pairs
| \( f(t) \) | \( F(s) \) |
|---|---|
| \( \displaystyle \delta(t) \), unit impulse at \(t=0\) | \( \displaystyle 1 \) |
| \( \displaystyle 1 \) or \( \displaystyle u_0(t) \), step at \(t=0\) | \( \displaystyle \frac{1}{s} \) |
| \( \displaystyle t \) or \( \displaystyle tu_0(t) \), ramp at \(t=0\) | \( \displaystyle \frac{1}{s^2} \) |
| \( \displaystyle e^{-\alpha t} \) | \( \displaystyle \frac{1}{s+\alpha} \) |
| \( \displaystyle t e^{-\alpha t} \) | \( \displaystyle \frac{1}{(s+\alpha)^2} \) |
| \( \displaystyle e^{-\alpha t}\sin(\beta t) \) | \( \displaystyle \frac{\beta}{(s+\alpha)^2+\beta^2} \) |
| \( \displaystyle e^{-\alpha t}\cos(\beta t) \) | \( \displaystyle \frac{s+\alpha}{(s+\alpha)^2+\beta^2} \) |
| \( \displaystyle \frac{d^n f(t)}{dt^n} \) | \( \displaystyle s^n F(s) - \sum_{m=0}^{n-1} s^{n-1-m}\frac{d^m f(0)}{dt^m} \) |
| \( \displaystyle \int_{0}^{t} f(\tau)\,d\tau \) | \( \displaystyle \frac{1}{s}F(s) \) |
| \( \displaystyle \int_{0}^{t} g(t-\tau)h(\tau)\,d\tau \) | \( \displaystyle G(s)H(s) \) |
| \( \displaystyle f(t-\tau) \) | \( \displaystyle e^{-\tau s}F(s) \) |