Mathematics - iii (Differential Calculus) - Vol 1 | 4. Second Shifting Theorem by Abraham | Learn Smarter
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4. Second Shifting Theorem

4. Second Shifting Theorem

The Laplace Transform is crucial for solving differential equations, especially when dealing with delayed functions through the Second Shifting Theorem. This theorem, using the Heaviside step function, enables transformations of functions that activate after a specified time. Its applications span various fields, illustrating its importance in analyzing real-world systems that exhibit delays.

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Sections

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  1. 1
    Laplace Transforms & Applications

    The Second Shifting Theorem in Laplace Transforms facilitates the handling...

  2. 1.1
    Second Shifting Theorem

    The Second Shifting Theorem in Laplace Transforms enables the handling of...

  3. 1.2
    Introduction

    The section introduces the Second Shifting Theorem and the importance of...

  4. 1.3
    Concept Of Heaviside Unit Step Function

    The Heaviside unit step function models functions that activate at a given...

  5. 1.4
    Second Shifting Theorem (Time Shifting In Laplace Domain)

    The Second Shifting Theorem simplifies the analysis of delayed functions in...

  6. 1.5
    Proof Of The Second Shifting Theorem

    The Second Shifting Theorem provides a method to transform delayed functions...

  7. 1.6
    Important Notes

    The Second Shifting Theorem is essential for handling delayed functions in...

  8. 1.7
    Graphical Representation

    The section discusses the Second Shifting Theorem, its application in...

  9. 1.8

    This section covers the Second Shifting Theorem in Laplace Transforms,...

  10. 1.9
    Applications Of Second Shifting Theorem

    The Second Shifting Theorem enables the Laplace Transform to handle delayed...

  11. 1.10

    The Second Shifting Theorem facilitates the transformation of time-delayed...

What we have learnt

  • The Laplace Transform is essential for transforming functions, especially in engineering and mathematics.
  • The Second Shifting Theorem allows for the transformation of delayed functions in the Laplace domain.
  • The unit step function is critical in modeling signals that start at a certain time.

Key Concepts

-- Laplace Transform
An integral transform used to convert a function of time into a function of a complex variable.
-- Second Shifting Theorem
A theorem stating that the Laplace transform of a delayed function can be expressed as an exponential factor times the transform of the original function.
-- Heaviside Step Function
A function that is zero for negative time values and one for positive, used to define delayed functions.

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