"A Fourier series is an expansion of a periodic function into a sum of trigonometric functions."
A way to express periodic functions as an infinite sum of sines and cosines.
Periodic Functions: Functions that repeat themselves over a fixed interval or period.
Trigonometric Functions: Functions based on the ratio of the sides of a right triangle, used extensively in Fourier series.
Fourier Series: A way to represent periodic functions as a sum of sine and cosine functions.
Orthogonality: Two functions are orthogonal if their inner product is zero. The sine and cosine functions are orthogonal to each other.
Inner Product: A mathematical operation that determines the extent to which two functions align with each other.
Convergence: The process of determining whether a Fourier series converges to the original function or not.
Complex Fourier Series: An alternative way to represent periodic functions using complex numbers rather than sine and cosine functions.
Fourier Transform: A mathematical representation of a non-periodic function as an infinite sum of sine and cosine waves.
Partial Differential Equations: Equations involving partial derivatives, which can be solved using Fourier series and transform.
Applications of Fourier Series: A variety of physical, engineering, and mathematical phenomena can be modeled using Fourier series, including heat transfer, music synthesis, and image processing.
Periodic Fourier Series: A periodic function can be expressed as an infinite sum of sines and cosines, with coefficients determined by integrating the product of the function and a specific sine or cosine over one period of the function.
Complex Fourier Series: A complex function can be expressed as a sum of complex exponential functions, using Fourier coefficients. The real and imaginary parts of the complex exponentials give the cosine and sine terms of the Fourier series, respectively.
Half-Range Fourier Series: A function on half the interval can be extended to the full interval by making it odd, resulting in a Fourier series that only consists of odd terms.
Trigonometric Fourier Series: Any periodic function can be expressed as a sum of cosines and sines of the fundamental frequency and its harmonics, using Fourier coefficients.
Fourier Transform: A continuous-time function is expressed as a sum of complex exponentials, where the coefficients are expressed as a function of frequency, which is the continuous analogue of Fourier series.
Discrete Fourier Transform: A finite sequence of numbers can be expressed as a sum of complex exponentials, with coefficients that can be calculated using the Fast Fourier transform algorithm.
Laplace Transform: A function of time can be transformed into a function of a complex variable, s, in the Laplace domain, which is the continuous analogue of Fourier transform.
"The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series."
"By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood."
"For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation."
"Fourier series cannot be used to approximate arbitrary functions because most functions have infinitely many terms in their Fourier series, and the series do not always converge."
"Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function."
"The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions."
"The study of the convergence of Fourier series focuses on the behaviors of the partial sums."
"Fourier series are closely related to the Fourier transform, which can be used to find the frequency information for functions that are not periodic."
"Periodic functions can be identified with functions on a circle, for this reason, Fourier series are the subject of Fourier analysis on a circle."
"The Fourier transform is also part of Fourier analysis, but is defined for functions on R^n."
"Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered."
"All of which are consistent with one another, but each of which emphasizes different aspects of the topic."
"Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time."
"Fourier analysis has birthed an area of mathematics called Fourier analysis."
"Fourier originally defined the Fourier series for real-valued functions of real arguments."
"Many other Fourier-related transforms have since been defined, extending his initial idea to many applications."
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