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Last Updated: Mar 27, 2024

Convolution Theorem For Laplace Transform

Author Prachi Singh
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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) * f(t)} = F1 (s) * F2 (s)

Where, f1 (t) * f(t) indicates convolution.



F (s) =  F1 (s) * F2 (s)


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


From the definition of Laplace Transform,

Hence, proved.

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Frequently Asked Questions

1. What is a Laplace transform?

A Laplace transform is an integral transform of a derivative function with a real variable ‘t’ which can be used to convert it into a complex function with a variable ‘s’. For t ≥ 0, let f(t) be given and assume the function satisfies certain conditions to be stated later on.

The Laplace transform of f(t), which is denoted by L{f(t)} or F(s) is defined by the Laplace transform formula:

2. Why do we use Laplace transform?

A Laplace transform is used generally for solving linear ordinary differential equations which involve the use of integral transforms. Differential equations can also be solved using a Laplace transform. 

It is widely accepted in many fields. A Laplace transform reduces linear differential equations (LDE) to an algebraic equation (relates calculus and algebra), which may then be solved using basic algebraic identities. This has 

a lot of applications in physics, electrical engineering, optics, control engineering, mathematics, signal processing, etc.

Key Takeaways

Congratulations on finishing the blog!! After reading this blog, you will grasp the Convolution Theorem for the Laplace Transform concept.

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