# Laplace transform of erf

1. Jan 8, 2006

### janrain

how do i find the laplace transform of the following error function without using tables?
f(t)=erf(t^(1/2))
i've been trying really long but i seem to be stuck in a loop of erf

2. Jan 8, 2006

### saltydog

Hey Jarain. Suppose you mean other than Mathematica right?

Just perform the integrations directly then:

$$\mathcal{L}\left\{\text{Erf}[\sqrt{t}]\right\}= \int_0^{\infty}e^{-st}\left[\frac{2}{\sqrt{\pi}}\int_0^{\sqrt{t}} e^{-u^2}du\right]dt$$

Now, can you switch the order of integrations to effect the solution?

Last edited: Jan 8, 2006
3. Jan 8, 2006

### Tide

Try integrating by parts.

4. Jan 8, 2006

### saltydog

Nice! Thanks.

Well, then do it both ways Jarain.

Edit: Oh yea. Tide's way is better.

5. Apr 20, 2011

### Calculusman08

Hi! I actually just performed this transform recently.
Let dv/dt = $$\int e^{-st}$$
Let u = $$\int_0^{\sqrt{t}} e^{-x^2} dx$$