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: [tex] {\cal L}^{-1} \{A\} [/tex]

where [itex] A = F(s) e^{C_2\sqrt{-s+C_1 }} [/itex]

and [itex] C_1,C_2 [/itex] are constants and [itex] F(s) [/itex] 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 [itex] F(s) [/itex]. I already applied boundary and initial conditions.


Thanks.

pebblesofsand

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

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

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

hunt_mat

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