Presentation on theme: "[YEAR OF ESTABLISHMENT – 1997]"— Presentation transcript:
1 [YEAR OF ESTABLISHMENT – 1997] DEPARTMENT OF MATHEMATICS[YEAR OF ESTABLISHMENT – 1997]DEPARTMENT OF MATHEMATICS, CVRCE
2 MATHEMATICS - II ● LAPLACE TRANSFORMS ● FOURIER SERIES FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING]● LAPLACE TRANSFORMS● FOURIER SERIES● FOURIER TRANSFORMS● VECTOR DIFFERENTIAL CALCULUS● VECTOR INTEGRAL CALCULUS● LINE, DOUBLE, SURFACE, VOLUME INTEGRALS● BETA AND GAMMA FUNCTIONSTEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8th EDITION]DEPARTMENT OF MATHEMATICS, CVRCE
3 FOURIER INTEGRAL [chapter – 10.8] MATHEMATICS - II LECTURE :16 DEPARTMENT OF MATHEMATICS, CVRCE
4 LAYOUT OF LECTURE FROM FOURIER SERIES TO FOURIER INTEGRAL INTRODUCTION & MOTIVATIONAPPLICATIONSEXISTENCE OF FOURIER INTEGRALSOME PROBLEMSFOURIER SINE AND COSINE INTEGRALSDEPARTMENT OF MATHEMATICS, CVRCE
5 INTRODUCTION & MOTIVATION FOURIER SERIES ARE POWERFUL TOOLS IN TREATING VARIOUS PROBLLEMS INVOLVING PERIODIC FUNCTIONS. HOWEVER, FOURIER SERIES ARE NOT APPLICABLE TO MANY PRACTICAL PROBLEMS SUCH AS A SINGLE PULSE OF AN ELECTRICAL SIGNAL OR MECHANICAL FORCE VIBRATION WHICH INVOLVE NONPERIODIC FUNCTIONS. THIS SHOWS THAT METHOD OF FOURIER SERIES NEEDS TO BE EXTENDED. HERE WE START WITH THE FOURIER SERIES OF AN ARBITRARY PERIODIC FUNCTION fL OF PERIOD 2L AND THEN LET L SO AS TO DEVELOP FOURIER INTEGRAL OF A NON-PERIODIC FUNCTION.Jean Baptiste Joseph Fourier (Mar21st 1768 –May16th 1830) French Mathematician & Physicist
6 FROM FOURIER SERIES TO FOURIER INTEGRAL Let fL (x) be an arbitrary periodic function whose period is 2L which can be represented by its Fourier series as follows:DEPARTMENT OF MATHEMATICS, CVRCE
7 FROM FOURIER SERIES TO FOURIER INTEGRAL From (1), (2), (3), and (4), we get
8 FROM FOURIER SERIES TO FOURIER INTEGRAL The representation (6) is valid for any fixed L, arbitrary large but finite. We now let L and assume that the resultant non-periodic function is absolutely integrable on x-axis , i.e. the resulting non-periodic function
9 FROM FOURIER SERIES TO FOURIER INTEGRAL is absolutely integrable.Set
10 FROM FOURIER SERIES TO FOURIER INTEGRAL Applying (8) and (9) in (7), we getRepresentation (10) with A(w) and B(w) given by (8) and (9), respectively, is called a Fourier integral of f(x).
12 EXISTENCE OF FOURIER INTEGRAL TheoremIf a function f(x) is piecewise continuous in every finite interval and has a right-hand derivative and left-hand derivative at every point and if the integralexists, then f(x) can be represented by a Fourier integral. At a point where f(x) is discontinuous the value of the Fourier integral equals the average of the left and right hand limits of f(x) at that point.
13 PROBLEMS ON FOURIER INTEGRAL EXAMPLE-1: Find the Fourier integral representation of the functionSolution:The Fourier integral of the given function f(x) is
17 DIRICHLET’S DISCONTINUOUS FACTOR From Example – 1 by Fourier integral representation, we find thatAt x= 1, the function f(x) is discontinuous. Hence the value of the Fourier integral at 1 is ½ [f(1-0)+f(1+0)] = ½[0+1]=1/2.
18 DIRICHLET’S DISCONTINUOUS FACTOR The above integral is called Dirichlet’s discontinuous factor.
19 Sine integral Dirichlet’s discontinuous factor is given by. Putting x= 0 in the above expression we get .The integral (1) is the limit of the integral as u The integral is called the sine integral and it is denoted by Si(u)