Fourier series Research, Essays (university) of Calculus for Engineers

Download Fourier series Research and more Essays (university) Calculus for Engineers in PDF only on Docsity! Fourier Series Research By Ali Albaghdadi Id: 20180007205 History of Fourier Research: Fourier series have made a huge impact on applied mathematics and theoretical. Fourier series are the sum of the orthonormal families which can approximate periodic, complex valued functions with arbitrary precision. There is a quite good number of methods for estimating complex numbers, but Fourier series are especially stand out because uniform convergence of the Fourier series. Furthermore, Fourier coefficients are made to minimize the error from the actual function. In mid eighteenth century, physical problems such as the conduction of patterns of heat and the study of oscillations and vibrations led to the study of Fourier series. The major issue that was faced is how arbitrary real valued functions could be presented by sum of simpler functions. Fourier series is an infinite sum of trigonometric functions that are used to model real live valued periodic functions. It has been used in signal processing and digital image processing for the analysis of a single image as a 2-D wave form, and many other types of similar quantum mechanics. Not only that but it also represents filter transformation, representation and encoding and much more. The Fourier breaks a function of time or signal into frequencies that makes in a way to how a musical chart can be broken into nodes and represented as pitches. It is also called the frequency domain representation of the original notes. Let S(x) be a periodic function with period [−π ,π ¿ an= 1 T ∫ −π π S (x ) cos (2 πx∗n2T )dx ,n≥0 bn= 1 T ∫ −π π S (x ) sin ⁡(2¿ πx∗n 2T )dx ,n≥0¿ S ( x )=a0+∑ n=1 ∞ (ancos nπxL +bn sin nπx L ) Leading to the formula we been using which is: ↓ S ( x )=∑ n=0 ∞ (an cos(nx)+bn sin(nx)) There are 4 types of Fourier: 1)A periodic continuous signal with a periodic spectrum. 2) Discrete a periodic spectrum and a periodic continuous signal. 3)Continuous periodic spectrum and periodic discrete signal. 4) periodic discrete signal and discrete periodic spectrum. Application of Fourier Series: Fourier series simplify the analysis of periodic, real valued functions, specifically, it can break up a periodic function into an infinite series of sine and cosines waves. This property makes Fourier series very helpful in many applications. Consider the common differential equation: f (t )=x ' ' (t )+a x ' (t )+b