Question

-

If L [F(t)] = F(s), then L [t" f(t)) =
​

Answer 1

Answer:

If L{f(t)} = F(s), then the inverse Laplace transform of F(s) is

L

−1

{F(s)} = f(t). (1)

The inverse transform L

−1

is a linear operator:

L

−1

{F(s) + G(s)} = L

−1

{F(s)} + L

−1

{G(s)}, (2)

and

L

−1

{cF(s)} = cL

−1

{F(s)}, (3)

for any constant c.

Answer 2

L [F(t)] = F(s)

Step-by-step explanation:

• Let f(t) be a function specified for t ≥ 0. Then, for every value of s where the integral converges, we define the Laplace transform of f(t) as the following function.

                           $F(S) = L{f(t)} = \int\limits^\alpha _0 {f(t)e^{-st} } \, dt$

• We translate a function from the t-domain to the s-domain by using the Laplace transform, where F(s) is a complex function of a complex variable. We're changing the problem into a domain that should be easier to address in this way.

• = $\int\limits^\alpha _0 {[af(t) + bg(t)]e^{-st} } \, dt$