Poles in the transformed fonction expose exponential in the original function - 3blue1brown

$\int_{0}^{\inf} f(t) \mathcal{e}^{st} \,dt$

$f(t) = 1$ -> $\int_{0}^{\inf} f(t) \mathcal{e}^{st} \,dt = 1/s$

Laplace transform is linear. It is very similar to the Fourrier transform (Laplace is a generalisation to non pure complex).

What does it means to integrate a complex function ?

• integrating scalar function gives area below the function
• but as generalisation it can be seen as the mean on sum of intervall
• this generalize as barycenter for complex function

Limited Continutation

• How to defined a complex function outside of it’s defintion domain
  • either there is no such extension
  • or there is only one that exist