Difference between two Laplace transform definitions

[goodness52200]

http://xs104.xs.to/xs104/06322/ssss.gif

Hello all ^^

What is the difference between the two definitions

thanks a lot

 

[WigneRacah]

In a heuristic contest, the first one takes account of distributions/measures (for istance "Dirac delta function" and its derivatives) with support in the origin. The second one doesn't.