Subject : Advance engineering mathematics Topic : Fourier series & Fourier integral

Fourier Series,Fourier Integral, Fourier Transform

Introduction Jean Baptiste Joseph Fourier (Mar21st 1768 –May16th 1830) French mathematician, physicist Main Work: (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 equations Proposal of his partial differential equation for conductive diffusion of heat Discovery of the "greenhouse effect“ Fourier series is very useful in solving ordinary and partial differential equation.

Even, Odd, and Periodic Functions Even, Odd, and Periodic Functions

Fourier Series of a Periodic Function Fourier Series of a Periodic Function Definition : A Fourier series may be defined as an expansion of a function in a series of sines and cosines such as, The coefficients are related to the periodic function f(x) by definite integrals: Henceforth we assume f satisfies the following (Dirichlet) conditions: (1) f(x) is a periodic function; (2) f(x) has only a finite number of finite discontinuities; (3) f(x) has only a finite number of extrem values, maxima and minima in the interval [0,2p].

EULER’S FORMULA The formula for a Fourier series is: N We have formulae for the coefficients (for the derivations see the course notes): One very important property of sines and cosines is their orthogonality, expressed by:

These formulae are used in the derivation of the formulae for Example – Find the coefficients for the Fourier series of: Find

Find, f (x) is an even function so:

Find Since both functions are even their product is even:

So we can put the coefficients back into the Fourier series formula:

Summary of finding coefficients function even function odd function neither Though maybe easy to find using geometry

Half range Expansions Half range Expansions It often happens in applications, especially when we solve partial differential equations by the method of separation of variables, that we need to expand a given function f in a Fourier series, where f is defined only on a finite interval. We define an “extended function”, say f ext, so that f ext is periodic in the domain of -∞< x < ∞, and f ext =f(x) on the original interval 0<x<L. There can be infinite number of such extensions. Four extensions: half- and quarter- range cosine and sine extensions, which are based on symmetry or antisymmetry about the endpoints x=0 and x=L.

HRC (half range cosines) f ext is symmetric about x=0 and also about x=L. Because of its symmetry about x=0, f ext is an even function, and its Fourier series will contain only cosines, no sines. Further, its period is 2L, so L is half the period.

HRS (half range sines)

Complex exponential form of Fourier series

PARSEVAL’S FORMULA (1)(1) (2)(2) If a function has a Fourier series given byFourier series then Bessel's inequality becomes an equality known as Parseval's theorem. From (1),Bessel's inequality1 Integrating so

Math for CSLecture 1120 Fourier Integral If f(x) and f’(x) are piecewise continuous in every finite interval, and f(x) is absolutely integrable on R, i.e. converges, then Remark: the above conditions are sufficient, but not necessary.

DIFFERENT FORMS OF FOURIER INTEGRAL THEOREM

Complex or exponential form

INFINITE FOURIER TRANSFORM

Fourier Sine Transform

Math for CSLecture 1125 Properties of Fourier transform 1Linearity: For any constants a, b the following equality holds: 2 Scaling: For any constant c, the following equality holds:

Math for CSLecture Time shifting: proof: 4. Frequency shifting: Proof:

Math for CSLecture Modulation: Proof: Using Euler formula, properties 1 (linearity) and 4 (frequency shifting):

Periodically forced oscillation: mass-spring system m = mass c = damping factor k = spring constant F(t) = 2L- periodic forcing function mx’’(t) + cx’(t) + k x(t) = F(t) Differential Equations for Engineers mF(t) k

The 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<t<0 1 if0<t<1

THANK YOU