Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Mathematical Physics ↓


Fourier Series


In mathematical physics, a Fourier series is a way of representing a function as a sum of sines and cosines. It helps analyze periodic functions by looking at them through their frequency components. With Fourier series, complex periodic phenomena can be broken down into simpler, oscillating components. This ability is particularly useful in fields such as acoustics, optics, electrical engineering, and quantum mechanics, where wave patterns are important.

Origin and significance

Fourier series are named after Jean-Baptiste Joseph Fourier, who introduced the idea while studying heat transfer problems in the early 19th century. Fourier's insight was that a periodic function could be expressed as an infinite sum of sines and cosines. This revelation was groundbreaking as it paved the way for developing a powerful analytical tool known as Fourier analysis.

Periodic functions

To understand Fourier series, we first need to understand what a recurring function is. A function ( f(t) ) is called recurring if there exists a positive number T such that for all values of t:

f(t + T) = f(t)

The smallest positive T for which this is true is called the period of the function.

The basic idea of Fourier series

The main idea behind Fourier series is that we can write any periodic function as a sum of simple sine and cosine functions. Mathematically, this is represented as:

f(t) = a_0 + sum_{n=1}^{infty} left[ a_n cosleft(frac{2pi nt}{T}right) + b_n sinleft(frac{2pi nt}{T}right) right]

Here, ( a_0 ), ( a_n ), and ( b_n ) are known as the Fourier coefficients.

Calculating the Fourier coefficients

To find the Fourier coefficients, we use the following formulas, which involve integration over one period of the function:

a_0 = frac{1}{T} int_{0}^{T} f(t) , dt a_n = frac{2}{T} int_{0}^{T} f(t) cosleft(frac{2pi nt}{T}right) , dt b_n = frac{2}{T} int_{0}^{T} f(t) sinleft(frac{2pi nt}{T}right) , dt

These coefficients measure how much of each sine and cosine is present in the function.

Visual example of Fourier series

Let's consider a simple example, let's say we want to approximate a square wave function. A square wave varies periodically between -1 and 1. Using Fourier series, it can be approximated by summing up several sine functions.

The more terms (sine) we include in the series, the closer we get to the actual square wave shape.

The blue line in the above SVG image shows the approximation of the square wave using Fourier series.

Text example of Fourier series

Consider a periodic serrated waveform defined as:

f(t) = t - lfloor t rfloor, ; 0 leq t < T

where ( lfloor t rfloor ) represents the floor function, or the largest integer less than or equal to ( t ). The Fourier series representation of this wave is:

f(t) = frac{1}{2} - sum_{n=1}^{infty} frac{1}{n} sinleft(frac{2pi nt}{T}right)

This represents the linear growth of the serrated wave, with the addition of numerous sine terms to match its pattern.

Even and odd functions

Functions can exhibit symmetry, which makes their Fourier series quite simple. A function is also called symmetric when:

f(-t) = f(t)

For even functions, all sign coefficients ( b_n ) are zero. Conversely, a function is odd if:

f(-t) = -f(t)

For odd functions, all cosine coefficients ( a_n ) (except possibly ( a_0 )) are zero. Understanding symmetry helps reduce complexity in calculations.

Practical applications

Fourier series have many applications. It is important to highlight some of these:

  1. Signal processing: Decomposes audio signals into individual frequencies, allowing specific sounds to be isolated or amplified.
  2. Image processing: Concise representation of images helps with tasks such as compression and edge detection.
  3. Vibrations and waves: Analysis of mechanical vibrations and frequency of sound waves in structures.

Convergence of Fourier series

An important aspect of any series is to understand its convergence, i.e., whether the infinite sum approaches a finite well-defined function. Fourier series converge under a wide range of conditions. For example, if a function ( f(t) ) is piecewise continuous and has a finite number of maximums and minimums within a period, then its Fourier series converges to ( f(t) ).

Drawbacks and the Gibbs phenomenon

Even though the Fourier series converges, it may exhibit a strange behavior near the discontinuity known as the Gibbs phenomenon. This appears as small oscillations near transition points in the function. Although these oscillations do not disappear with more terms in the series, they become negligible in many practical cases.

Variations and expansions

Based on the Fourier series concept, several other analytical methods are available:

  • Fourier transform: A generalization for converting functions from the time to the frequency domain, even for non-periodic functions.
  • Laplace transform: Often used for control system analysis because of its uniform treatment of marginal conditions.
  • Discrete Fourier Transform (DFT): Used for analyzing discrete signals, usually implemented in its faster version - the Fast Fourier Transform (FFT).

Conclusion

Fourier series stand as a pillar of mathematical physics, enabling complex functions to be investigated in simpler terms. The ability to break down a function into sinusoidal components provides insights that underlie many modern technological marvels and theoretical advances in science and engineering.

By capturing the essence of periodicity and frequency, Fourier's work endures as an invaluable tool across many disciplines, continually demonstrating the profound power of mathematical abstraction.


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Fourier Series

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