Problem with finding an Inverse Laplace Transform

[pebblesofsand]

I've been messing around with Laplace transforms. Anyway to get to the point I arrived at a "solution" in the s domain and got stuck.

I'm trying to solve for the inverse laplace transform of A: $${\cal L}^{-1} \{A\}$$

where $A = F(s) e^{C_2\sqrt{-s+C_1 }}$

and $C_1,C_2$ are constants and $F(s)$ is a function of s.

Is there any way to apply the shifting theorem to this equation? If not how do I go about solving the above? I don't know much about $F(s)$. I already applied boundary and initial conditions.


Thanks.

 

[pebblesofsand]

I guess I should clarify that I am trying to find ${\cal L}^{-1} \{A\}=a(x,t)$ for any $F(s)$.

In the above equation $C_2$ is $x$. So the equation is actually $A = F(s) e^{x\sqrt{-s+C_1 }}$. I wrote $C_2$ in the place of $x$ because I was trying to look up the transform in tables.

I had specified the boundary condition as $a(-\infty,t)=f(t)$ This transforms to the $s$ domain as $A(-\infty,s)=F(s)$. Is there any way to get a general form for $a(x,t)$ for any $F(s)$ or would I have to specify $F(s)$?

 

[hunt_mat]

I think that the way that you're going to have to tackle this is residue calculus.