Fourier series-Mathematics

Fourier series

1. Fourier series with period 2L

2. Fourier series of even and odd function

3. Fourier sine and cosine

A Fourier series represents a periodic function as an infinite sum of sines and cosines. It is widely used in signal processing, engineering, physics, and mathematics to analyze periodic functions.

The integrals determine how much of each sine and cosine wave contributes to reconstructing f(x).

Applications of Fourier Series

1. Signal Processing – Used to analyze and reconstruct signals.

2. Electrical Engineering – Helps in circuit analysis and AC waveforms.

3. Physics – Used in heat transfer, quantum mechanics, and wave equations.

4. Vibration Analysis – Helps analyze mechanical systems.

   
   

   Fourier Series for Even and Odd Functions

   When working with Fourier series, recognizing whether a function is even or odd simplifies calculations by determining which terms in the series will be present. Effect on Fourier Series

   • Since even functions are symmetric about the y-axis, their Fourier series contain only cosine terms (no sine terms).

     Key Takeaways

     1. Even function → Only cosine terms

     2. Odd function → Only sine terms

     3. Mixed function → Both sine & cosine terms

        Fourier Sine and Cosine Series

        The Fourier sine and cosine series are special cases of the Fourier series when a function is either odd or even on a given interval.