# A laplace transform

1. Mar 19, 2007

### catcherintherye

I am trying to show that L[t^m] = m!/s^m+1, unfortunately I can not understand why integral from 0 to inf of (t^m)(e^-st)dt = (-d/ds)^m. integral 0 to inf of e^-stdt...... ????????

2. Mar 19, 2007

### marcusl

You can exchange the order, that is, take -d/ds inside the integral, since they are both linear operations. It takes down a factor t from the exponential. Do it m times and get t^m.

3. Mar 20, 2007

### HallsofIvy

Staff Emeritus
If m is a positive integer, use induction together with integration by parts:
$$\int_0^\infty t^m e^{-st}dt$$
Let u= tm, dv= e-stdt. Then du= m tm-1 and v= -1/s e-st. The integral becomes
$$-\frac{m}{s}\int_0^\infty t^{m-1}e^{-st}dt$$
since uv= 0 at both ends.