7.2 Fourier Series

Detailed Theory: Fourier Series

1. Introduction to Fourier Series

1.1 What is a Fourier Series?

A Fourier series is a way to represent a periodic function as a sum of simple sine and cosine waves.

Key Idea: Any periodic function (repeating pattern) can be broken down into a sum of:

• Sine waves

• Cosine waves

• Constant term

1.2 Why Use Fourier Series?

1. Simplify complex waves into basic components

2. Analyze signals in engineering and physics

3. Solve differential equations more easily

4. Filter signals in electronics

1.3 Basic Components

A Fourier series has three types of terms:

1. Constant term (average value)

2. Cosine terms (even symmetry)

3. Sine terms (odd symmetry)

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2. Periodic Functions

2.1 Definition

A function $f(x)$ is periodic with period $T$ if:

$$f(x + T) = f(x) \quad \text{for all } x$$

Examples:

• Sine wave: $\sin(x)$ has period $2\pi$

• Square wave: Repeats every $T$

• Sawtooth wave: Repeats every $T$

2.2 Common Periods

1. Fundamental period: Smallest $T > 0$ for which $f(x+T) = f(x)$

2. Angular frequency: $\omega = \frac{2\pi}{T}$

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3. Fourier Series Representation

3.1 General Form

For a function $f(x)$ with period $2L$, the Fourier series is:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]$$

3.2 Fourier Coefficients

The coefficients are calculated as:

Constant term:

$$a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$$

Cosine coefficients:

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

Sine coefficients:

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

3.3 Special Case: Period $2\pi$

If $f(x)$ has period $2\pi$ ($L = \pi$):

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(nx) + b_n \sin(nx) \right]$$

With coefficients:

$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx$$

$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx$$

$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx$$

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4. Symmetry and Fourier Series

4.1 Even Functions

Definition: $f(-x) = f(x)$ for all $x$

Properties for Fourier series:

• Cosine terms only

• $b_n = 0$ for all $n$

• Simplified formulas:

$$a_0 = \frac{2}{L} \int_{0}^{L} f(x) \, dx$$

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

Examples: $\cos(x)$, $x^2$, $|x|$

4.2 Odd Functions

Definition: $f(-x) = -f(x)$ for all $x$

Properties for Fourier series:

• Sine terms only

• $a_n = 0$ for all $n$

• Simplified formulas:

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

Examples: $\sin(x)$, $x$, $x^3$

4.3 Neither Even nor Odd

If function has no symmetry:

• Both sine and cosine terms needed

• Use full formulas

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5. Common Fourier Series Examples

5.1 Square Wave

Function: Period $2\pi$

$$f(x) = \begin{cases} -1 & \text{if } -\pi < x < 0 \\ 1 & \text{if } 0 < x < \pi \end{cases}$$

Fourier series (odd function):

$$f(x) = \frac{4}{\pi} \left[ \sin(x) + \frac{\sin(3x)}{3} + \frac{\sin(5x)}{5} + \cdots \right]$$

$$f(x) = \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\sin((2n-1)x)}{2n-1}$$

5.2 Sawtooth Wave

Function: $f(x) = x$ for $-\pi < x < \pi$, period $2\pi$

Fourier series (odd function):

$$f(x) = 2 \left[ \sin(x) - \frac{\sin(2x)}{2} + \frac{\sin(3x)}{3} - \frac{\sin(4x)}{4} + \cdots \right]$$

$$f(x) = 2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \sin(nx)}{n}$$

5.3 Triangle Wave

Function: Even function, period $2\pi$

Fourier series:

$$f(x) = \frac{\pi}{2} - \frac{4}{\pi} \left[ \cos(x) + \frac{\cos(3x)}{9} + \frac{\cos(5x)}{25} + \cdots \right]$$

$$f(x) = \frac{\pi}{2} - \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\cos((2n-1)x)}{(2n-1)^2}$$

5.4 Half-Wave Rectifier

Function: $f(x) = \sin(x)$ for $0 < x < \pi$, $0$ for $\pi < x < 2\pi$

Fourier series:

$$f(x) = \frac{1}{\pi} + \frac{1}{2} \sin(x) - \frac{2}{\pi} \left[ \frac{\cos(2x)}{3} + \frac{\cos(4x)}{15} + \frac{\cos(6x)}{35} + \cdots \right]$$

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6. Convergence of Fourier Series

6.1 Dirichlet Conditions

For Fourier series to converge to $f(x)$, the function must satisfy:

1. Periodic: $f(x+T) = f(x)$

2. Finite number of discontinuities in one period

3. Finite number of maxima/minima in one period

4. Absolutely integrable: $\int_{-L}^{L} |f(x)| \, dx < \infty$

6.2 Pointwise Convergence

At points where $f(x)$ is continuous:

$$\text{Fourier series} \to f(x)$$

At points where $f(x)$ has a jump discontinuity:

$$\text{Fourier series} \to \frac{f(x^+) + f(x^-)}{2}$$

where $f(x^+)$ is right-hand limit, $f(x^-)$ is left-hand limit.

6.3 Gibbs Phenomenon

Overshoot near discontinuities in partial sums.

Important: Even with more terms, overshoot doesn't disappear (about 9% of jump).

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7. Complex Form of Fourier Series

7.1 Euler's Formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

$$\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$$

$$\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$$

7.2 Complex Fourier Series

For period $2L$:

$$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n\pi x / L}$$

Complex coefficients:

$$c_n = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i n\pi x / L} \, dx$$

Relation to real coefficients:

$$c_0 = \frac{a_0}{2}$$

$$c_n = \frac{a_n - ib_n}{2} \quad \text{for } n > 0$$

$$c_{-n} = \frac{a_n + ib_n}{2} \quad \text{for } n > 0$$

Advantages:

• More compact notation

• Easier for some calculations

• Useful in advanced applications

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8. Parseval's Theorem

8.1 Energy Interpretation

Parseval's theorem relates the average power of a function to its Fourier coefficients.

8.2 Theorem Statement

For Fourier series of $f(x)$ with period $2L$:

$$\frac{1}{2L} \int_{-L}^{L} [f(x)]^2 \, dx = \left(\frac{a_0}{2}\right)^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$$

In complex form:

$$\frac{1}{2L} \int_{-L}^{L} |f(x)|^2 \, dx = \sum_{n=-\infty}^{\infty} |c_n|^2$$

8.3 Applications

1. Signal power calculation

2. Checking Fourier series accuracy

3. Summing certain infinite series

Example: For square wave:

$$\frac{1}{\pi} \int_{-\pi}^{\pi} [f(x)]^2 \, dx = 1 = \frac{8}{\pi^2} \left(1 + \frac{1}{9} + \frac{1}{25} + \cdots \right)$$

This gives: $\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} = \frac{\pi^2}{8}$

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9. Fourier Series on Different Intervals

9.1 General Interval [a, b]

If function defined on $[a, b]$ with $b-a = 2L$:

Change of variable: Let $t = \frac{\pi(x-a)}{L}$

Then Fourier series becomes:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi(x-a)}{L}\right) + b_n \sin\left(\frac{n\pi(x-a)}{L}\right) \right]$$

Coefficients:

$$a_n = \frac{2}{b-a} \int_{a}^{b} f(x) \cos\left(\frac{2n\pi(x-a)}{b-a}\right) \, dx$$

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

9.2 Half-Range Expansions

When function defined only on $[0, L]$:

a) Fourier Cosine Series (even extension)

Extend as even function on $[-L, L]$:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$$

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

b) Fourier Sine Series (odd extension)

Extend as odd function on $[-L, L]$:

$$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$$

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

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10. Applications of Fourier Series

10.1 Signal Processing

1. Filter design: Remove unwanted frequencies

2. Audio compression: MP3, AAC formats

3. Image compression: JPEG format

10.2 Solving Differential Equations

Heat equation: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$

Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Method:

1. Assume solution as Fourier series

2. Substitute into equation

3. Solve for coefficients

10.3 Electrical Engineering

1. AC circuit analysis: Non-sinusoidal voltages/currents

2. Power calculations: For non-sinusoidal waveforms

3. Harmonic analysis: Identify frequency components

10.4 Physics

1. Quantum mechanics: Wave functions

2. Optics: Diffraction patterns

3. Vibrations: Mechanical systems

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11. Important Formulas Summary

11.1 Fourier Series (Period 2L)

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]$$

$$a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$$

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

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

11.2 Even Functions

$$b_n = 0$$

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

11.3 Odd Functions

$$a_n = 0$$

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

11.4 Complex Form

$$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n\pi x / L}$$

$$c_n = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i n\pi x / L} \, dx$$

11.5 Parseval's Theorem

$$\frac{1}{2L} \int_{-L}^{L} [f(x)]^2 \, dx = \left(\frac{a_0}{2}\right)^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$$

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12. Solved Examples

Example 1: Basic Fourier Series

Find Fourier series for $f(x) = x^2$ on $[-\pi, \pi]$

Solution: Since $f(x)$ is even, $b_n = 0$

$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \, dx = \frac{2}{\pi} \int_{0}^{\pi} x^2 \, dx = \frac{2}{\pi} \cdot \frac{\pi^3}{3} = \frac{2\pi^2}{3}$$

$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) \, dx = \frac{2}{\pi} \int_{0}^{\pi} x^2 \cos(nx) \, dx$$

Integration by parts gives:

$$a_n = \frac{4(-1)^n}{n^2}$$

Thus:

$$f(x) = \frac{\pi^2}{3} + 4 \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos(nx)$$

Example 2: Square Wave

Find Fourier series for:

$$f(x) = \begin{cases} 0 & \text{if } -\pi < x < 0 \\ 1 & \text{if } 0 < x < \pi \end{cases}$$

Solution:

$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx = \frac{1}{\pi} \int_{0}^{\pi} 1 \, dx = 1$$

$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx = \frac{1}{\pi} \int_{0}^{\pi} \cos(nx) \, dx = 0$$

$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx = \frac{1}{\pi} \int_{0}^{\pi} \sin(nx) \, dx$$

$$b_n = \frac{1}{\pi} \left[ -\frac{\cos(nx)}{n} \right]_{0}^{\pi} = \frac{1 - (-1)^n}{n\pi}$$

Thus:

$$f(x) = \frac{1}{2} + \frac{2}{\pi} \left[ \sin(x) + \frac{\sin(3x)}{3} + \frac{\sin(5x)}{5} + \cdots \right]$$

Example 3: Using Symmetry

Find Fourier cosine series for $f(x) = x$ on $[0, \pi]$

Solution: Even extension to $[-\pi, \pi]$:

$$a_0 = \frac{2}{\pi} \int_{0}^{\pi} x \, dx = \frac{2}{\pi} \cdot \frac{\pi^2}{2} = \pi$$

$$a_n = \frac{2}{\pi} \int_{0}^{\pi} x \cos(nx) \, dx$$

Integration gives:

$$a_n = \frac{2[(-1)^n - 1]}{\pi n^2}$$

Thus:

$$f(x) = \frac{\pi}{2} - \frac{4}{\pi} \left[ \cos(x) + \frac{\cos(3x)}{9} + \frac{\cos(5x)}{25} + \cdots \right]$$

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13. Exam Tips and Common Mistakes

13.1 Common Mistakes

1. Wrong interval: Check period and integration limits

2. Ignoring symmetry: Always check if function is even/odd

3. Constant term: $a_0$ divided by 2 in series

4. Integration errors: Double-check integration by parts

5. Missing n=0 case: Treat $a_0$ separately

13.2 Problem-Solving Strategy

1. Determine period of function

2. Check symmetry (even/odd/neither)

3. Choose appropriate formulas based on symmetry

4. Compute coefficients carefully

5. Write final series in simplified form

13.3 Quick Checks

1. Even function: Only cosine terms

2. Odd function: Only sine terms

3. Average value: $a_0/2$ is DC component

4. Convergence: At discontinuities, series converges to average of left/right limits

5. Parseval's theorem: Useful for verifying calculations

This comprehensive theory covers all essential aspects of Fourier series with practical examples and formulas, providing complete preparation for the entrance examination.