I have probably helped all I can without some real understanding of what you are actually trying to do. I really think you need to post your real problem that you are trying to solve. Or, you can try all options and then pick the solution that makes physics sense given the problem you are trying to solve. We can constuct any piecewise constant function by adding together step functions shifted in. With these standard #F(s)# examples, in the original problem formulation you should probably be able to deduce the region of convergence. Sympy can calculate laplace transforms of the step easily. Of course, this example has an #F(s)# that is analytic everywhere in the complex #s# plane except at the 2 poles, so falls under that standard types of functions for which we use the Bromwich integral approach. Wait - in your example above you have #f(t)# that is not zero for #t1# then #f(t) = u(t) (e^ + e^t)#. I am starting to think that your #F(s)# probably cannot represent the Laplace transform of either a regular or generalized function, but I may be wrong at this point. Indeed, any #F(s)# that is zero for a right-half plane has an inverse transform that is zero (for a reference see Chapter 6 of "Mathematics for the Physical Sciences" by Schwartz, hopefully in a library you have access to). It asks for two functions and its intervals. So your Bromwich integral must be along a vertical contour with #\Re(s)>1# your function is zero there, so the answer you get is zero. Laplace transform for Piecewise functions Added by sam.st in Mathematics Widget for the laplace transformation of a piecewise function. Exercise Find the Laplace Transform of the piecewise function f(t). If we blindly assume that your Laplace transform is valid, for your case it is analytic for #\Re(s)>1# (since it is not analytic for, say #\Re(s)>1/2#). Laplace Transforms of Piecewise Continuous Functions. I have never ever worked with piecewise Laplace transforms (am not even sure they can possibly represent a valid Laplace transform). (I may ask the moderators to combine these threads as they are highly related), the Laplace transform will be analytic in a right-half of the complex plane. Clearly for a finite k, the kth order derivative of F exists for all s except 1, but how about as k -> Inf?įinally, just wondering if the two conditions I listed initially (the two limits) are sufficient for the inverse Laplace transform of $F(s)$ to exist. I'm also not sure whether Post's inversion formula can be used since I'm not sure I understand how to evaluate high-order derivatives of a function which is not differentiable at s = 1. I'm not sure whether the Bromwich integral method can be applied, since it would appear that if I choose gamma (the Browmich integral integration limits: gamma - i*Inf to gamma + i*Inf) between 0 and 1 the function to integrate is (1-s), whereas if I choose gamma > 1 then the Bromwich integral is obviously 0. I understand the conditions for the existence of the inverse Laplace transforms areĬlearly the limits above do satisfy the existence of the inverse condition, but I'm not sure how to determine the inverse.
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