Mathematics - iii (Differential Calculus) - Vol 1 | 13. Convolution Theorem by Abraham | Learn Smarter
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13. Convolution Theorem

The Convolution Theorem is a critical concept in Laplace Transforms that facilitates the inverse transformation of product functions. It is defined through a unique operation on piecewise continuous functions and encompasses significant properties such as commutativity, associativity, and distributivity. The theorem finds application across various domains, including differential equations and signal processing, providing a powerful tool for engineers to manage complex systems.

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Sections

  • 13

    Laplace Transforms & Applications

    The Laplace Transform is a crucial tool in engineering and mathematics, particularly for solving linear differential equations, with the Convolution Theorem simplifying inverse transformations.

    Learning Practice

  • 13.1

    Convolution Theorem

    The Convolution Theorem provides a method to simplify the inverse Laplace transform of a product of functions in the s-domain.

    Learning Practice

  • 13.1.1

    Introduction

    This section introduces the Convolution Theorem within the context of Laplace Transforms, emphasizing its significance in simplifying the inverse Laplace transform of products of functions.

    Learning Practice

  • 13.1.2

    Definition Of Convolution

    The Convolution Theorem simplifies the inverse Laplace transformation of products of functions by facilitating operations in the time domain.

    Learning Practice

  • 13.1.3

    Convolution Theorem (Statement)

    The Convolution Theorem simplifies the process of finding the inverse Laplace Transform of the product of two Laplace transforms through convolution.

    Learning Practice

  • 13.1.4

    Proof Of Convolution Theorem (Sketch)

    The Convolution Theorem establishes a crucial link between the Laplace transforms of functions and their convolution in the time domain.

    Learning Practice

  • 13.1.5

    Properties Of Convolution

    The section introduces the Convolution Theorem, highlighting its definition, properties, and applications in solving problems related to Laplace Transforms.

    Learning Practice

  • 13.1.5.1

    Commutative

    This section explores the Commutative property of convolution within the context of the Laplace Transform.

    Learning Practice

  • 13.1.5.2

    Associative

    The Convolution Theorem provides a method to compute the inverse Laplace transform of a product of two functions via integration of their convoluted forms.

    Learning Practice

  • 13.1.5.3

    Distributive Over Addition

    The section examines the Distributive Property of convolution over addition, outlining its mathematical formulation and significance in the context of Laplace Transforms.

    Learning Practice

  • 13.1.6

    Applications

    The Convolution Theorem simplifies the process of finding the inverse Laplace transform of products of functions.

    Learning Practice

  • 13.1.7

    Solved Examples

    This section presents solved examples that illustrate the application of the Convolution Theorem in Laplace Transforms.

    Learning Practice

  • 13.1.8

    Graphical Interpretation

    The Convolution Theorem simplifies the inverse Laplace transform of a product of Laplace functions by defining convolution in the time domain.

    Learning Practice

  • 13.1.9

    Summary

    The Convolution Theorem simplifies the inverse Laplace transform of the product of two functions, allowing for easier analysis in various engineering applications.

    Learning Practice

References

Unit 1 ch13.pdf

Class Notes

Memorization

What we have learnt

  • The Convolution Theorem sim...
  • The convolution of two func...
  • The theorem is characterize...

Final Test

Revision Tests