Unit 4 - Notes

MTH174 7 min read

Unit 4: Fourier Series

1. Introduction to Fourier Series

A Fourier series is an expansion of a periodic function into an infinite sum of sines and cosines. In engineering mathematics, Fourier series are essential tools for analyzing periodic phenomena, such as vibrations, alternating currents, and heat conduction.

Periodic Functions

A function $f(x)$ is said to be periodic with period $T$ if $f(x + T) = f(x)$ for all $x$ in the domain of $f$ . The fundamental period is the smallest positive value of $T$ for which this holds. Standard trigonometric functions like $\sin(x)$ and $\cos(x)$ are periodic with period $2\pi$ .

A Fourier series allows us to represent a general periodic function $f(x)$ of period $2\pi$ as:
$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)]$
where $a_0$ , $a_n$ , and $b_n$ are constants known as the Fourier coefficients.

2. Euler's Formulae

To find the Fourier coefficients $a_0$ , $a_n$ , and $b_n$ for a function $f(x)$ defined over an interval of length $2\pi$ (such as $[-\pi, \pi]$ or $[0, 2\pi]$ ), we use Euler's formulae. These are derived using the orthogonality properties of sine and cosine functions.

For an interval $[c, c+2\pi]$ , the coefficients are calculated as follows:

For $a_0$ :
$a_0 = \frac{1}{\pi} \int_{c}^{c+2\pi} f(x) \, dx$
(Note: Some texts define the first term as $a_0$ instead of $a_0/2$ . If defined as $a_0/2$ , the formula above holds. This represents twice the average value of the function over one period).
For $a_n$ :
$a_n = \frac{1}{\pi} \int_{c}^{c+2\pi} f(x) \cos(nx) \, dx \quad \text{for } n = 1, 2, 3, \dots$
For $b_n$ :
$b_n = \frac{1}{\pi} \int_{c}^{c+2\pi} f(x) \sin(nx) \, dx \quad \text{for } n = 1, 2, 3, \dots$

Orthogonality Relations

Euler's formulae depend on the fact that the trigonometric system is orthogonal over any interval of length $2\pi$ . For any integers $m \neq n$ :

$\int_{-\pi}^{\pi} \cos(mx) \cos(nx) \, dx = 0$
$\int_{-\pi}^{\pi} \sin(mx) \sin(nx) \, dx = 0$
$\int_{-\pi}^{\pi} \sin(mx) \cos(nx) \, dx = 0$

3. Conditions for a Fourier Expansion (Dirichlet's Conditions)

Not every periodic function can be expanded into a Fourier series. For a Fourier series to converge to the function $f(x)$ , the function must satisfy a set of sufficient conditions known as Dirichlet's Conditions.

If $f(x)$ is defined and periodic with period $2\pi$ , its Fourier series converges to $f(x)$ provided that over one period:

$f(x)$ is single-valued and finite.
$f(x)$ has at most a finite number of finite discontinuities (it is piecewise continuous).
$f(x)$ has at most a finite number of maxima and minima (bounded variation).

Note: Dirichlet's conditions are sufficient but not strictly necessary. However, they cover almost all functions encountered in practical engineering problems.

4. Functions Having Points of Discontinuity

When a function satisfies Dirichlet's conditions, its Fourier series converges to $f(x)$ at all points where $f(x)$ is continuous.

However, at a point of jump discontinuity, say $x = x_0$ , the Fourier series does not converge to either the left-hand limit or the right-hand limit. Instead, it converges to the arithmetic mean (average) of the left-hand and right-hand limits at that point.

Mathematically, at a point of discontinuity $x_0$ , the sum of the Fourier series $S(x_0)$ is:
$S(x_0) = \frac{f(x_0^+) + f(x_0^-)}{2}$
where:

$f(x_0^+) = \lim_{h \to 0^+} f(x_0 + h)$ (Right-hand limit)
$f(x_0^-) = \lim_{h \to 0^+} f(x_0 - h)$ (Left-hand limit)

5. Change of Interval

In many practical applications, periodic functions do not have a period of $2\pi$ . Instead, they have an arbitrary period $T = 2L$ . We can adapt the Fourier series to a function defined over an interval $[-L, L]$ or $[c, c+2L]$ by making a change of variables.

Let $z = \frac{\pi x}{L}$ . Then, as $x$ varies from $-L$ to $L$ , $z$ varies from $-\pi$ to $\pi$ .

The Fourier series for a function $f(x)$ with period $2L$ is:
$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]$

The modified Euler's formulae for the interval $[-L, L]$ become:

$a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$
$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$
$b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

(If the interval is $[0, 2L]$ , the limits of integration change to $0$ to $2L$ , but the multiplier $\frac{1}{L}$ remains the same).

6. Even and Odd Functions

Recognizing whether a function is even or odd can significantly reduce the computational effort required to find the Fourier coefficients, as certain coefficients automatically become zero.

Even Functions

A function is even if $f(-x) = f(x)$ for all $x$ .

Geometrically, the graph of an even function is symmetrical about the y-axis.
Examples: $\cos(x)$ , $x^2$ , $|x|$ .
Fourier Coefficients for Even Functions over $[-L, L]$ :
Because the product of an even function and a sine function (which is odd) is an odd function, the integral over a symmetric interval is zero. Therefore, $b_n = 0$ .
$a_0 = \frac{2}{L} \int_{0}^{L} f(x) \, dx$
$a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$
The resulting series is called a Fourier Cosine Series.

Odd Functions

A function is odd if $f(-x) = -f(x)$ for all $x$ .

Geometrically, the graph of an odd function is symmetrical about the origin.
Examples: $\sin(x)$ , $x^3$ , $x$ .
Fourier Coefficients for Odd Functions over $[-L, L]$ :
Because the product of an odd function and a cosine function (even) is odd, the integrals for $a_0$ and $a_n$ are zero. Therefore, $a_0 = 0$ and $a_n = 0$ .
$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$
The resulting series is called a Fourier Sine Series.

7. Half-Range Series

Sometimes, a function is only defined over a finite interval $[0, L]$ , and we need to represent it as a Fourier series. We can do this by artificially extending the function to the interval $[-L, 0]$ to make it either strictly even or strictly odd. This creates a "half-range" series.

Half-Range Cosine Series

To obtain a cosine series, we extend $f(x)$ as an even function in $[-L, 0]$ . Since it is even, $b_n = 0$ .
The series is:
$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$
Where:

$a_0 = \frac{2}{L} \int_{0}^{L} f(x) \, dx$
$a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$

Half-Range Sine Series

To obtain a sine series, we extend $f(x)$ as an odd function in $[-L, 0]$ . Since it is odd, $a_0 = 0$ and $a_n = 0$ .
The series is:
$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$
Where:

$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

Application: Half-range series are especially useful in solving partial differential equations (like the heat and wave equations) where boundary conditions naturally dictate the use of only sines or only cosines.

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