| \( f(t) \) | \( F(s) \) |
| 1 | \( \dfrac{1}{s} \) |
| \( t \) | \( \dfrac{1}{s^2} \) |
| \( t^n \) | \( \dfrac{n!}{s^{n+1}} \), \( (n = 1,2,3...) \) |
| \( t^{1/2} \) | \( \dfrac{\sqrt{\pi}}{2s^{3/2}} \) |
| \( t^{-1/2} \) | \( \sqrt{\dfrac{\pi}{s}} \) |
| \( e^{-at} \) | \( \dfrac{1}{s + a} \) |
| \( t e^{-at} \) | \( \dfrac{1}{(s + a)^2} \) |
| \( \sin(at) \) | \( \dfrac{a}{s^2 + a^2} \) |
| \( t \sin(at) \) | \( \dfrac{2as}{(s^2 + a^2)^2} \) |
| \( e^{-at} \sin(bt) \) | \( \dfrac{b}{(s + a)^2 + b^2} \) |
| \( \cos(at) \) | \( \dfrac{s}{s^2 + a^2} \) |
| \( t \cos(at) \) | \( \dfrac{s^2 - a^2}{(s^2 + a^2)^2} \) |
| \( e^{-at} \cos(bt) \) | \( \dfrac{s + a}{(s + a)^2 + b^2} \) |
| \( \sinh(at) \) | \( \dfrac{a}{s^2 - a^2} \) |
| \( \cosh(at) \) | \( \dfrac{s}{s^2 - a^2} \) |
| \( 1 - \cos(at) \) | \( \dfrac{a^2}{s(s^2 + a^2)} \) |
| \( \left(\dfrac{2}{t}\right)(t - \cos(at)) \) | \( \ln \left( \dfrac{s^2 + a^2}{s^2} \right) \) |
| \( \left(\dfrac{2}{t}\right)(t - \cosh(at)) \) | \( \ln \left( \dfrac{s^2 - a^2}{s^2} \right) \) |
| \( \left(\dfrac{1}{t}\right)(\sin(at)) \) | \( \arctan \left( \dfrac{a}{s} \right) \) |