Mathematics - iii (Differential Calculus) - Vol 1 | 12. Inverse Laplace Transform by Abraham | Learn Smarter
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12. Inverse Laplace Transform

The Inverse Laplace Transform is essential for retrieving time-domain functions from their Laplace-transformed equivalents. Several methods, including partial fractions, convolution, and the Complex Inversion Formula, facilitate this transformation. Its applications span various fields such as electrical engineering, control systems, and mechanical systems, particularly in solving ordinary differential equations.

Sections

  • 12

    Laplace Transforms & Applications

    The Inverse Laplace Transform is a crucial mathematical tool to retrieve time-domain functions from their Laplace-transformed versions.

    Learning Practice

  • 12.1

    Inverse Laplace Transform

    The Inverse Laplace Transform retrieves time-domain functions from their Laplace transforms, crucial for solving differential equations.

    Learning Practice

  • 12.2

    Definition

    The Inverse Laplace Transform retrieves time-domain functions from their corresponding Laplace transforms.

    Learning Practice

  • 12.3

    Basic Inverse Laplace Transforms

    The Inverse Laplace Transform retrieves time-domain functions from their Laplace-transformed expressions, playing a crucial role in various fields such as engineering and mathematics.

    Learning Practice

  • 12.3.1

    Partial Fraction Method

    The Partial Fraction Method is an essential technique used to decompose rational functions in order to facilitate the calculation of inverse Laplace transforms.

    Learning Practice

  • 12.3.2

    Convolution Theorem

    The Convolution Theorem provides a method for finding the inverse Laplace transform of the product of two Laplace transforms.

    Learning Practice

  • 12.3.3

    Complex Inversion Formula (Bromwich Integral)

    The Complex Inversion Formula, expressed through the Bromwich Integral, is a theoretical approach used in the inverse Laplace transform for retrieving time-domain functions from the complex frequency domain.

    Learning Practice

  • 12.3.4

    Heaviside’s Expansion Formula (For Distinct Poles)

    This section introduces Heaviside's Expansion Formula for finding the inverse Laplace transform of rational functions with distinct linear factors.

    Learning Practice

  • 12.4

    Properties Of Inverse Laplace Transform

    The properties of the Inverse Laplace Transform facilitate the retrieval of time-domain functions from their Laplace transforms using specific properties.

    Learning Practice

  • 12.5

    Applications Of Inverse Laplace Transform

    The Inverse Laplace Transform retrieves time-domain functions from Laplace transforms, proving essential in various fields, such as differential equations and control systems.

    Learning Practice

  • 12.6

    Practice Problems

    This section contains practice problems that focus on the methods and applications of the Inverse Laplace Transform.

    Learning Practice

  • 12.7

    Summary

    The Inverse Laplace Transform is a technique for converting Laplace transforms back to their original time-domain functions, critical in engineering and applied mathematics.

    Learning Practice

References

Unit 1 ch12.pdf

Class Notes

Memorization

What we have learnt

  • The Inverse Laplace Transfo...
  • Techniques include partial ...
  • Common in solving different...

Final Test

Revision Tests