Introduction
According to the Convolution Theorem for Laplace transform,
Given: f1 (t) and f2 (t) as the two Laplace transformable functions.
F1 (s) is the Laplace transform of f1 (t) and F2 (s) is the Laplace transform of f2 (t).
To Prove: The product of F1 (s) and F2 (s) is the Laplace transform of f(t) which is the result of the convolution of f1 (t) and f2 (t).
The resultant convolution is denoted by f1 (t) * f2 (t) and is calculated further using the equation:
L {f1 (t) * f2 (t)} = F1 (s) * F2 (s)
Where, f1 (t) * f2 (t) indicates convolution.
Proof
Let
F (s) = F1 (s) * F2 (s)
Where,
x and y are dummy variables.
As x and y are independent variables we can write it as follows.
Consider new variable ‘t’ such that
Now,
From the definition of Laplace Transform,
Hence, proved.
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