Posts Tagged ‘Fourier series’
Thursday, September 24th, 2009
Introduction to the Fourier Series
Description
A detailed tutorial on the solving of Fourier series. Step by step tutorial including several examples of how to solve the Fourier series for reference.
Overview
The Fourier series is very common in differential equations and partial differential equations. What a Fourier series does is decompose a periodic function into the sum of simple oscillating functions, like sine and cosine. Fourier series are part of Fourier analysis and were first introduced by Joseph Fourier to solve the heat equation. So another name for the Fourier series is the heat equation, although they are considered different things. Both the heat equation and the Fourier series are partial differential equations.
Tags: cosine, decompose, differential equations, Fourier analysis, Fourier series, function, functions, heat equation, Joseph Fourier, Math, oscillating, partial differential equations, periodic function, simple, sine, sum
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Thursday, September 24th, 2009
Dirichlet Kernel Explained
Description
A detailed tutorial on the solving of the Dirichlet kernel. Step by step tutorial including several examples of how to solve the Dirichlet kernel for reference.
Overview
The Dirichlet kernel is this collection of functions: 
The Dirichlet kernel is important because of its realtionship with the Fourier series. Other noteable things about the Dirichlet kernel is that it uses the Delta function, and also it uses a trigonometric identity.
Tags: collection, convolution, Delta, differential equations, Dirichlet kernel, Fourier series, function, Johann Peter Gustav Lejeune Dirichlet, Math, trigonometric identity
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Tuesday, September 22nd, 2009
How to Solve Parseval’s Theorem
Description
A detailed tutorial on the solving of Parseval’s Theorem. Step by step tutorial including several examples of how to solve Parseval’s Theorem for reference.
Overview
Parseval’s Theorem is sometimes called Rayleigh’s Energy Theorem, or Rayleigh’s Identity. The term is used to describe the unitarity of any Fourier’s transform, but the most general form of this property should be referred to as the Plancherel Theorem.
Parseval’s Theorem states: Suppose that A(X) and B(x) are two Riemann integrable, complex-valued functions on R of period 2pi with Fourier series 
and 
respectively. Then 
where horizontal bars indicate complex conjugation.
Tags: complex, complex conjugation, Fourier series, function, integrable, Math, Parseval's Theorem, Plancherel theorem, Rayleigh's energy theorem, Rayleigh's Identity, Riemann, unitarity, unitary
Posted in Differential Equations | No Comments »
Tuesday, September 22nd, 2009
Gibbs Phenomenon Explained
Description
A detailed tutorial and explanation of the Gibbs Phenomenon. Step by step tutorial including several examples of the Gibbs Phenomenon for reference.
Overview
Gibbs Phenomenon refers to the way the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. What happens is the nth partial sum of the series has large oscillations near the jump, which could possibly overshoot the partial sum above the function. However, the overshoot does not die out as the frequency increases – it approaches a finite limit.
Tags: finite limit, Fourier series, frequency, function, Gibbs, Gibbs Phenomenon, J. Willard Gibbs, jump, jump discontinuity, Math, oscillation, overshoot, partial sum, periodic function, piecewise, square wave
Posted in Differential Equations | No Comments »
