Unit-4: Fourier Series

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1. Introduction and Euler's Formulae

Fourier series represents a periodic function as a sum of sines and cosines. It is a powerful tool used in mathematical analysis, particularly in solving differential equations, signal processing, and analyzing waveforms. By expressing complex periodic functions as an infinite series of trigonometric functions, Fourier series makes it easier to analyze and manipulate them.

Euler's formulae are essential in finding the Fourier coefficients, which determine the weights of the sine and cosine terms in the series. The coefficients are calculated using the following formulas:

a₀ = (1/T) ∫_-T/2^T/2 f(x) dx   (Represents the average value of the function over one period)

a_n = (2/T) ∫_-T/2^T/2 f(x) cos(nωx) dx   (Coefficient for the cosine terms)

b_n = (2/T) ∫_-T/2^T/2 f(x) sin(nωx) dx   (Coefficient for the sine terms)

Where T is the period of the function and ω = 2π/T is the angular frequency.

2. Conditions for Fourier Expansion

A function f(x) can be expanded as a Fourier series if the following conditions, called Dirichlet's conditions, are satisfied:

• f(x) is periodic with period T, meaning f(x + T) = f(x).
• f(x) is piecewise continuous, i.e., it may have a finite number of discontinuities, but it must be bounded.
• f(x) has a finite number of maxima and minima within one period.

3. Functions with Points of Discontinuity

At a point of discontinuity, the Fourier series does not exactly match the function value. Instead, it converges to the average of the left-hand limit and the right-hand limit at that point. This is expressed as:

f(c) = (f(c+) + f(c-)) / 2

This phenomenon is called the Gibbs phenomenon, where oscillations occur near discontinuities, but they diminish as more terms are added to the series.

4. Change of Interval

Often, functions are not defined on the standard interval [-π, π]. In such cases, the interval can be scaled and adjusted to match the desired interval [a, b]. The transformed variable is given by:

x = (b - a)/2 * u + (a + b)/2

This scaling allows the Fourier series to be applied to a variety of practical problems.

5. Even and Odd Functions

A function f(x) is classified as:

• Even Function: f(x) = f(-x), symmetric about the y-axis. For even functions, all the sine coefficients (b_n) are zero because sine is an odd function.
• Odd Function: f(x) = -f(-x), symmetric about the origin. For odd functions, all the cosine coefficients (a_n) are zero because cosine is an even function.

This property simplifies Fourier series calculations.

6. Half Range Series

When a function is defined only on a limited interval [0, L], we can extend it to form a half-range Fourier series. There are two common extensions:

• Half-Range Sine Series: Extend the function as an odd function to obtain only sine terms.
• Half-Range Cosine Series: Extend the function as an even function to obtain only cosine terms.

This method is useful in solving boundary value problems in engineering and physics.