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.