Comparing $e^{-s}$ to the transform pairs, equation 6 looks the best place to start. The inverse Laplace transform of the function Y(s) is the unique function y(t) that is continuous on [0,infty) and satisfies L[y(t)](s)=Y(s). From this it follows that we can have two different functions with the same Laplace transform. k is a function having an inverse Laplace transform. I need to find the inverse Laplace transform of the following function: $$F(s) = \frac{(s-2)e^{-s}}{s^2-4s+3}$$ I completed the square on the bottom and got the following: The formula for Inverse Laplace transform is; How to Calculate Laplace Transform? Let’s now use the linearity to compute a few inverse transforms.! This inverse laplace transform can be found using the laplace transform table [1]. A consequence of this fact is that if L[F(t)] = f(s) then also L[F(t) + N(t)] = f(s). Laplace transform table. To use this in computing our desired inverse transform… TABLE OF LAPLACE TRANSFORM FORMULAS L[tn] = n! The Laplace transform is defined with the L{} operator: Inverse Laplace transform. s n+1 L−1 1 s = 1 (n−1)! Function name Time domain function tn−1 L eat = 1 s−a L−1 1 s−a = eat L[sinat] = a s 2+a L−1 1 s +a2 = 1 a sinat L[cosat] = s s 2+a L−1 s s 2+a = cosat Diﬀerentiation and integration L d dt f(t) = sL[f(t)]−f(0) L d2t dt2 f(t) = s2L[f(t)]−sf(0)−f0(0) L dn … The … The Inverse Laplace Transform Calculator helps in finding the Inverse Laplace Transform Calculator of the given function. If all possible functions y(t) are discontinous one can select a piecewise continuous function to be the inverse transform. Example: Compute the inverse Laplace transform q(t) of Q(s) = 3s (s2 +1)2 You could compute q(t) by partial fractions, but there’s a less tedious way. Laplace transform makes the equations simpler to handle. 7. Find the inverse Laplace Transform of the function F(s). When a higher order differential equation is given, Laplace transform is applied to it which converts the equation into an algebraic equation, thus making it easier to handle. So the Inverse Laplace transform is given by: g(t)=1/3cos 3t*u(t-pi/2) The graph of the function (showing that the switch is turned on at t=pi/2 ~~ 1.5708) is as follows: Laplace transform function. Uniqueness of inverse Laplace transforms. Remember, L-1 [Y(b)](a) is a function that y(a) that L(y(a) )= Y(b). The inverse Laplace transform can be calculated directly. However, we see from the table of Laplace transforms that the inverse transform of the second fraction on the right of Equation \ref{eq:8.2.14} will be a linear combination of the inverse transforms $e^{-t}\cos t\quad\mbox{ and }\quad e^{-t}\sin t \nonumber$ The Laplace transform of a null function N(t) is zero. Example 26.3: Let’s ﬁnd L−1 1 s2 +9 t. We know (or found in table 24.1 on page 484) that L−1 3 s2 +9 t = sin(3t) , which is almost what we want. Solution: For the fraction shown below, the order of the numerator polynomial is not less than that of the denominator polynomial, therefore we first perform long division . Usually the inverse transform is given from the transforms table. Now we can express the fraction as a … Example. So, generally, we use this property of linearity of Laplace transform to find the Inverse Laplace transform.