[88869] in Discussion of MIT-community interests
How To Play The Piano
daemon@ATHENA.MIT.EDU (Piano Lessons Books)
Wed Sep 21 10:10:15 2016
Date: Wed, 21 Sep 2016 16:10:10 +0200
From: "Piano Lessons Books" <contact@piano15.info>
Reply-To: "Learn Jazz Piano" <contact@piano15.info>
To: <mit-talk-mtg@charon.mit.edu>
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<body><a href="http://piano15.info/vwltQroa6Acu1CdAElPL8c4b4KI57LcwvEK9ZbQ"><img border="0" src="http://piano15.info/pr8hohgY6LZ5_Tfs0DexGJRoQTy1bpDc8OHNmO0" /> </a>
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<p style="color: #32424E; margin: 12px; font: 11px sans-serif;">Can't browse our Adver_tisement below due to no images? <a href="http://piano15.info/vZKUa3fnk-LG22HCXxBNmRTvj0zLrFLGaxJtSa8" target="_blank"> Be sure to browse this,</a></p>
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<center><a href="http://piano15.info/vZKUa3fnk-LG22HCXxBNmRTvj0zLrFLGaxJtSa8" style=" font: 24px Arial Narrow; color: #713e00 ; margin: 15px auto;" target="_blank"><strong>Now ANYONE Can Learn Piano or Keyboard</strong></a></center>
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<strong><font color="#2C475A" size="+1">Imagine being able to sit down at a piano and just PLAY </font></strong><br />
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<p><strong><font color="#2C475A">Pianoforall is specially designed to take complete<br />
Beginners to an intermediate level faster than any other method.</font></strong></p>
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<font color="#E6E6E6" face="arial,helvetica,sans-serif" size="-1">In mathematics, a Fourier series (English pronunciation: /ˈfɔərieɪ/) is a way to represent a (wave-like) function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time FoThe Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.[nb 1] Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles. The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration urier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1. Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem. The study of Fourier series is a branch of Fourier analysis. Contents </font><br />
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