Mathematics (Civil Engineering -1) | 9. Fourier Integrals by Abraham | Learn Smarter
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9. Fourier Integrals

Fourier Integrals are essential for handling non-periodic functions in engineering applications, particularly in Civil Engineering contexts. This chapter discusses the derivation of Fourier Integrals, their formulas, applications in various engineering problems, and how they differ from Fourier Series. Key concepts include the properties of the Fourier Transform and the significance of Fourier Integrals in solving heat conduction and vibration problems, making it critical for understanding transient phenomena.

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Sections

  • 9

    Fourier Integrals

    Fourier Integrals extend the concept of Fourier Series to represent non-periodic functions as a continuous sum of sine and cosine functions, crucial for engineering applications.

    Learning Practice

  • 9.1

    The Need For Fourier Integrals

    Fourier Integrals are essential for expressing non-periodic functions through continuous superposition, allowing for practical applications in engineering.

    Learning Practice

  • 9.2

    Derivation Of The Fourier Integral

    This section discusses the derivation of the Fourier Integral, which allows the representation of non-periodic functions as a continuous superposition of sine and cosine functions.

    Learning Practice

  • 9.3

    Fourier Integral Formula

    The Fourier Integral Formula provides a method to express non-periodic functions as a continuous series of sines and cosines, essential for applications in engineering.

    Learning Practice

  • 9.4

    Fourier Cosine And Sine Integrals

    This section describes the Fourier integrals specific to even and odd functions, providing essential tools for solving problems in symmetric domains.

    Learning Practice

  • 9.5

    Conditions For Fourier Integrability

    This section outlines the necessary conditions for a function to possess a valid Fourier Integral representation.

    Learning Practice

  • 9.6

    Applications In Civil Engineering

    Fourier integrals are crucial tools in civil engineering applications, addressing non-periodic phenomena such as heat conduction and dynamic load analysis.

    Learning Practice

  • 9.7

    Worked Examples

    This section presents worked examples illustrating the application of Fourier integrals for functions expressed through sine and cosine representations.

    Learning Practice

  • 9.8

    Dirichlet’s Integral

    Dirichlet's Integral provides essential results used in Fourier integrals, particularly important in boundary value problems.

    Learning Practice

  • 9.9

    Complex Form Of Fourier Integral

    This section discusses the complex form of Fourier integrals, which provides a more elegant representation of functions using exponential functions.

    Learning Practice

  • 9.10

    Properties Of The Fourier Transform

    This section discusses the key properties of the Fourier Transform, which are crucial for solving various problems in engineering and applied mathematics.

    Learning Practice

  • 9.11

    Fourier Integral In Engineering Problem Solving

    The section discusses the application of Fourier integrals in engineering, particularly in solving heat diffusion problems using the Fourier transform.

    Learning Practice

  • 9.12

    Parseval’s Theorem For Fourier Integrals

    Parseval's theorem establishes a relationship between the time domain and frequency domain representations of a signal, particularly focusing on energy.

    Learning Practice

  • 9.13

    Comparison: Fourier Series Vs Fourier Integral

    Fourier Series and Fourier Integrals are compared to highlight their applicability for periodic and non-periodic functions respectively.

    Learning Practice

  • 9.14

    Common Integral Forms For Reference

    This section presents common integral forms used in Fourier analysis, specifically tailored for engineering applications.

    Learning Practice

  • 9.15

    Exercises

    This section presents exercises related to the application of Fourier integrals in various contexts.

    Learning Practice

References

Chapter_9_Fourie.pdf

Class Notes

Memorization

What we have learnt

  • Fourier Integrals extend th...
  • A function must be piecewis...
  • Fourier Integrals are used ...

Final Test

Revision Tests