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Fourier Series ,Fourier Integral, Fourier TransformGroup #9

Yizhi HongJiaqi Zhang Nicholas ZentaySagar Lonkar

17.1 IntroductionJean Baptiste Joseph Fourier (Mar21st 1768 May16th 1830) French mathematician, physicistMain Work:Thorie analytique de la chaleur (The Analytic Theory of Heat)Any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable (Incorrect)The concept of dimensional homogeneity in equationsProposal of his partial differential equation for conductive diffusion of heat

Discovery of the "greenhouse effect"http://en.wikipedia.org/wiki/Joseph_Fourier17.2 Even, Odd, and Periodic FunctionsExamples:Even function : Cosine function i.e. cos()

Odd function: Sine function i.e. sin()

Periodic Function: Both sine and cosine functions are periodic with a period of 2

17.3 Fourier Series of a Periodic FunctionFor FS f to represent f we need the series to converge, and we need its sum function to be the same as the original function f(x).THEOREM 17.3.1 Fourier Convergence Theroem

Proof of Euler FormulasExample: The function given by: y=-x - x0y=x 0x

The period of the above function is 2.Thus 2l = 2 Therefore l=

s0s1s3s5Example:Periodically forced oscillation: mass-spring systemm = massc = damping factork = spring constantF(t) = 2L- periodic forcing functionmx(t) + cx(t) + k x(t) = F(t)

http://www.jirka.org/diffyqs/Differential Equations for Engineers mF(t)kThe particular solution xp of the above equation is periodic with the same period as F(t) .The coefficients are k=2, and m=1 and c=0 (for simplicity). The units are the mks units (meters-kilograms- seconds). There is a jetpack strapped to the mass, which fires with a force of 1 newton for 1 second and then is off for 1 second, and so on. We want to find the steady periodic solution.

The equation is:x + 2x = F(t) Where F(t) => 0 if -1