Glossary

11.1 Fourier Series Expansion

5 min read•july 30, 2024

Expansion is a powerful tool for breaking down complex periodic functions into simpler parts. It's like taking a complicated song and splitting it into individual notes. This method helps us understand and work with tricky functions in math and engineering.

By representing functions as sums of sines and cosines, we can solve tough problems in physics and engineering. It's super useful for analyzing things like sound waves, electrical signals, and heat transfer. Fourier Series Expansion is a key player in making sense of the world around us.

Representing periodic functions with Fourier series

Fourier series representation

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  • Fourier series is a method of representing periodic functions as an infinite sum of sine and cosine functions with different frequencies and amplitudes
  • A repeats its values at regular intervals, satisfying the condition f(x)=f(x+T)f(x) = f(x + T) for all xx, where TT is the period
  • The Fourier series representation of a periodic function f(x)f(x) with period 2π2π is given by:
    • f(x)=a02+∑n=1∞[ancos⁡(nx)+bnsin⁡(nx)]f(x) = \frac{a₀}{2} + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)]
    • a0a₀, ana_n, and bnb_n are
    • nn represents the
  • The Fourier series allows for the decomposition of complex periodic functions into a sum of simple harmonic components, which can be analyzed and manipulated individually

Applicability and conditions

  • The Fourier series representation is applicable to both continuous and discontinuous periodic functions, as long as the function satisfies certain conditions ()
    • The function is periodic with period 2π2π
    • The function is piecewise continuous on the interval [−π,π][-π, π]
    • The function has a finite number of maxima and minima on the interval [−π,π][-π, π]
  • Examples of periodic functions that can be represented by Fourier series include:
  • Fourier series can be used to approximate non-periodic functions over a finite interval by extending the function periodically outside the interval

Fourier series coefficients

Calculation of Fourier coefficients

  • The Fourier coefficients a0a₀, ana_n, and bnb_n can be calculated using the following formulas:
    • a0=1π∫−ππf(x)dxa₀ = \frac{1}{π} \int_{-π}^{π} f(x) dx
    • an=1π∫−ππf(x)cos⁡(nx)dxa_n = \frac{1}{π} \int_{-π}^{π} f(x) \cos(nx) dx
    • bn=1π∫−ππf(x)sin⁡(nx)dxb_n = \frac{1}{π} \int_{-π}^{π} f(x) \sin(nx) dx
  • To determine the Fourier coefficients, the given periodic function f(x)f(x) is multiplied by the corresponding trigonometric function (11, cos⁡(nx)\cos(nx), or sin⁡(nx)\sin(nx)) and integrated over one period
  • The resulting integrals are evaluated using appropriate integration techniques, such as trigonometric substitution, integration by parts, or using known integral formulas

Piecewise defined functions

  • In cases where the periodic function is defined piecewise, the Fourier coefficients are calculated by splitting the integral into subintervals corresponding to each piece of the function
  • Example: For a square wave defined as f(x)={1,0<x<π−1,−π<x<0f(x) = \begin{cases} 1, & 0 < x < π \\ -1, & -π < x < 0 \end{cases}
    • The Fourier coefficients are calculated separately for each subinterval [0,π][0, π] and [−π,0][-π, 0]
    • The results are then combined to obtain the final Fourier coefficients
  • The calculated Fourier coefficients can be substituted back into the Fourier series representation to obtain the complete Fourier series expansion of the periodic function

Fourier series convergence

Types of convergence

  • of a Fourier series refers to whether the series approaches the original function as the number of terms in the series increases
  • occurs when the Fourier series converges to the function value at each point where the function is continuous
    • Example: The Fourier series of a continuous function converges pointwise to the function at every point in its domain
  • occurs when the maximum difference between the function and its Fourier series approximation approaches zero as the number of terms increases, for all points in the domain
    • Example: The Fourier series of a smooth, continuous function converges uniformly to the function over its entire domain

Gibbs phenomenon

  • is observed when a Fourier series approximates a function with discontinuities, resulting in oscillations near the discontinuities that do not diminish as the number of terms increases
  • The oscillations occur because the Fourier series tries to approximate the discontinuity with a sum of continuous functions
  • Example: The Fourier series approximation of a square wave exhibits Gibbs phenomenon at the discontinuities, with overshoots and undershoots that do not disappear as more terms are added to the series
  • Convergence of a Fourier series has implications for the accuracy of the approximation and the smoothness of the approximated function

Fourier series for boundary value problems

Solving PDEs with Fourier series

  • Fourier series can be used to solve (BVPs) in (PDEs), such as the , , and
  • The general steps to solve a BVP using Fourier series are:
    1. Formulate the PDE and specify the boundary conditions and initial conditions (if applicable)
    2. Assume a solution in the form of a Fourier series with unknown coefficients
    3. Substitute the assumed solution into the PDE and boundary conditions to obtain a system of equations for the Fourier coefficients
    4. Solve the system of equations to determine the Fourier coefficients
    5. Substitute the obtained Fourier coefficients back into the assumed solution to get the final solution of the BVP
  • Fourier series are particularly useful for solving BVPs in rectangular domains with homogeneous boundary conditions (Dirichlet, Neumann, or mixed)

Choosing sine or cosine functions

  • The choice of sine or cosine functions in the Fourier series depends on the type of boundary conditions imposed on the problem
  • For Dirichlet boundary conditions (specified function values at the boundaries), the Fourier series typically involves sine functions
    • Example: A vibrating string with fixed ends can be modeled using a Fourier sine series
  • For Neumann boundary conditions (specified derivatives at the boundaries), the Fourier series typically involves cosine functions
    • Example: Heat transfer in a rod with insulated ends can be modeled using a Fourier cosine series
  • Fourier series solutions provide an analytical representation of the solution, allowing for further analysis and interpretation of the physical phenomenon described by the PDE

Key Terms to Review (25)

Boundary Value Problems: Boundary value problems are mathematical problems in which one seeks to find a solution to a differential equation that satisfies certain conditions at the boundaries of the domain. These problems arise frequently in physics and engineering, especially in the context of heat conduction, vibrations, and fluid flow, as they help model various physical systems and their behavior under specific constraints.
Complex fourier series: A complex Fourier series is a way to express a periodic function as a sum of complex exponentials. This method uses Euler's formula to rewrite the trigonometric functions in terms of exponentials, allowing for more elegant mathematical manipulations and simplifications. By using complex coefficients, the series can capture both the amplitude and phase information of the original function, making it a powerful tool in the analysis of periodic signals.
Convergence: Convergence refers to the property of a sequence or function where it approaches a specific value as it progresses toward a limit. In the context of transforms and series, convergence is crucial as it determines whether these mathematical representations can accurately describe a system or signal across time and frequency domains. Understanding convergence is essential for ensuring that calculations yield meaningful results and that the transformations employed behave predictably.
Cosine function: The cosine function is a fundamental trigonometric function defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. It plays a critical role in representing periodic phenomena, making it essential in the analysis of waveforms and oscillations, particularly in Fourier series expansions where it helps decompose complex periodic signals into simpler components.
Dirichlet Conditions: Dirichlet conditions are a set of mathematical criteria that a function must satisfy to ensure that its Fourier series converges to the function itself at points within its interval. These conditions help determine the legitimacy of representing functions as Fourier series, especially for periodic functions, making them a cornerstone in the analysis of signal processing and vibrations.
Fourier Coefficients: Fourier coefficients are the numerical values that represent the amplitudes of the sine and cosine functions in a Fourier series expansion. These coefficients are crucial because they allow us to break down complex periodic functions into simpler trigonometric components, making analysis and computation much easier. By calculating these coefficients, one can reconstruct the original function from its Fourier series representation, highlighting its frequency components.
Fourier Series: A Fourier series is a way to represent a periodic function as a sum of simple sine and cosine functions. This powerful mathematical tool breaks down complex signals into their fundamental frequency components, making it easier to analyze and understand their behavior. Fourier series connect to various concepts in dynamic systems by providing insights into both homogeneous and non-homogeneous solutions, frequency response, and the transformations between time and frequency domains.
Fourier Transform: The Fourier Transform is a mathematical operation that transforms a time-domain signal into its frequency-domain representation, allowing us to analyze the frequency components present in the signal. This transformation provides insight into how different frequencies contribute to the overall signal, making it an essential tool for various applications in engineering and physics.
Gibbs Phenomenon: The Gibbs Phenomenon refers to the peculiar behavior of Fourier series when they approximate a discontinuous function. Specifically, it describes how, near a jump discontinuity, the Fourier series overshoots the actual value of the function, resulting in an oscillation that never fully diminishes, regardless of how many terms are included in the series. This phenomenon highlights the limitations of Fourier series in accurately representing functions with sharp transitions.
Harmonic number: A harmonic number, denoted as $H_n$, is the sum of the reciprocals of the first n natural numbers, represented mathematically as $H_n = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. This concept is pivotal in analyzing Fourier series expansions, especially when determining convergence properties and approximating functions with sine and cosine series.
Heat equation: The heat equation is a partial differential equation that describes how heat diffuses through a given region over time. It models the distribution of temperature in a medium as a function of both space and time, typically represented as $$u_t = eta u_{xx}$$, where $$u$$ is the temperature, $$t$$ is time, and $$x$$ is the spatial variable. This equation is essential in mathematical physics and engineering, particularly when dealing with problems of thermal conduction.
Inverse fourier transform: The inverse Fourier transform is a mathematical operation that converts frequency domain data back into the time domain, allowing us to retrieve the original signal from its frequency representation. This operation is crucial for analyzing signals and systems, as it provides a way to understand how different frequency components combine to form the overall signal. In the context of Fourier series expansion, the inverse Fourier transform helps to reconstruct periodic signals using their sinusoidal components.
Laplace's Equation: Laplace's equation is a second-order partial differential equation given by the formula $$ abla^2 ext{u} = 0$$, where $$ abla^2$$ is the Laplacian operator and $$ ext{u}$$ is a scalar function. This equation plays a critical role in various fields such as physics and engineering, particularly in studying steady-state heat distribution, electrostatics, and fluid flow. Solutions to Laplace's equation are harmonic functions, which means they satisfy the mean value property and exhibit unique properties that are essential for solving boundary value problems.
Orthogonality: Orthogonality refers to the property of being perpendicular or independent in a certain context, often in relation to functions or vectors. In the realm of Fourier Series Expansion, orthogonal functions are crucial because they allow for the decomposition of complex periodic signals into simpler, non-overlapping sine and cosine components. This independence ensures that each frequency component can be analyzed and manipulated without interference from others, simplifying the representation of signals.
Parseval's Theorem: Parseval's Theorem states that the total energy of a signal in the time domain is equal to the total energy of its representation in the frequency domain. This important principle highlights the relationship between time and frequency analyses, affirming that the sum of the squares of a signal's amplitude is conserved whether viewed in the time domain or transformed using techniques like Fourier series or Fourier transforms.
Partial Differential Equations: Partial differential equations (PDEs) are mathematical equations that involve unknown multivariable functions and their partial derivatives. They are crucial for describing various phenomena in physics, engineering, and applied mathematics, capturing how changes in one variable affect others. PDEs can model systems with multiple dimensions, making them essential for dynamic modeling and signal processing.
Periodic Function: A periodic function is a function that repeats its values at regular intervals, known as its period. This means that for any value of the variable, the function returns to the same output after a certain distance or time. Periodic functions are fundamental in mathematics and physics, especially when analyzing waveforms and oscillations, where they describe phenomena that cycle over time.
Pointwise Convergence: Pointwise convergence is a type of convergence for sequences of functions, where a sequence of functions converges at each point in the domain. This means that for every point in the domain, as you go further along in the sequence, the values of the functions approach a specific limit. In the context of function series, such as Fourier series expansion, pointwise convergence helps in understanding how well the series represents a function at specific points.
Sawtooth wave: A sawtooth wave is a non-sinusoidal waveform that rises linearly and then sharply drops, resembling the teeth of a saw. This waveform is characterized by its linear ascent and abrupt descent, making it distinct from other waveforms like sine and square waves. Sawtooth waves are important in various applications such as audio synthesis, signal processing, and Fourier series expansion due to their rich harmonic content.
Sine function: The sine function is a mathematical function that describes the relationship between an angle and the ratios of the lengths of the sides of a right triangle. It is fundamental in trigonometry and plays a key role in various applications, including waveforms and oscillations. In the context of Fourier series expansion, the sine function helps in representing periodic functions as sums of sine and cosine terms, allowing for the analysis of complex signals.
Square wave: A square wave is a non-sinusoidal waveform that alternates between a high and low state at a constant frequency, creating a waveform that looks like a series of squares. This waveform is significant in signal processing and communications because it can be used to represent digital signals, where the high state represents '1' and the low state represents '0'. Its distinct harmonic content makes it relevant when discussing Fourier series expansions, as it can be represented as a sum of sine waves of various frequencies.
Triangle wave: A triangle wave is a non-sinusoidal waveform that oscillates between a minimum and maximum value in a linear fashion, creating a triangular shape when plotted over time. This waveform is characterized by its linear rise and fall, with equal time spent on the upward and downward slopes. Triangle waves are often used in signal processing and can be analyzed using Fourier series expansion to understand their harmonic content.
Trigonometric series: A trigonometric series is a type of mathematical series that expresses a function as a sum of sine and cosine functions. These series play a crucial role in Fourier analysis, enabling the representation of periodic functions through their harmonic components. By breaking down complex waveforms into simpler trigonometric terms, trigonometric series provide insights into the frequency content and other properties of signals.
Uniform convergence: Uniform convergence refers to a type of convergence for a sequence of functions where the speed of convergence is uniform across the entire domain. This means that, for any given level of accuracy, there exists a single index beyond which all functions in the sequence differ from the limit function by less than that level, regardless of the point in the domain. This concept is important because it ensures the preservation of certain properties, such as continuity and integrability, when passing to the limit in function sequences, especially in the context of Fourier series expansion.
Wave equation: The wave equation is a second-order linear partial differential equation that describes the propagation of waves, such as sound waves, light waves, and water waves. It is fundamental in physics and engineering because it models how waves travel through different media, showing relationships between displacement, velocity, and acceleration over time and space.
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