DEPARTMENT OF MATHEMATICS YEAR OF ESTABLISHMENT 1997 DEPARTMENT

DEPARTMENT OF MATHEMATICS [YEAR OF ESTABLISHMENT – 1997] DEPARTMENT OF MATHEMATICS, CVRCE

MATHEMATICS - II ● LAPLACE FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING] TRANSFORMS ● FOURIER SERIES ● FOURIER TRANSFORMS ● VECTOR DIFFERENTIAL CALCULUS ● VECTOR INTEGRAL CALCULUS ● LINE, DOUBLE, SURFACE, TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION] DEPARTMENT OF MATHEMATICS, CVRCE

MATHEMATICS - II LECTURE : 16 FOURIER INTEGRAL [chapter – 10. 8] DEPARTMENT OF MATHEMATICS, CVRCE

LAYOUT OF LECTURE INTRODUCTI ON & MOTIVATION FROM FOURIER SERIES TO FOURIER INTEGRAL APPLICATIONS EXISTENCE OF FOURIER INTEGRAL FOURIER SINE AND COSINE INTEGRALS SOME PROBLEMS DEPARTMENT OF MATHEMATICS, CVRCE

INTRODUCTION & MOTIVATION Jean Baptiste Joseph Fourier (Mar 21 st 1768 –May 16 th 1830) French Mathematician & Physicist 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 f. L OF PERIOD 2 L AND THEN LET L SO AS TO DEVELOP FOURIER INTEGRAL OF A NON-PERIODIC FUNCTION.

FROM FOURIER SERIES TO FOURIER INTEGRAL Let f. L (x) be an arbitrary periodic function whose period is 2 L which can be represented by its Fourier series as follows: DEPARTMENT OF MATHEMATICS, CVRCE

FROM FOURIER SERIES TO FOURIER INTEGRAL From (1), (2), (3), and (4), we get

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 nonperiodic function

FROM FOURIER SERIES TO FOURIER INTEGRAL is absolutely integrable. Set

FROM FOURIER SERIES TO FOURIER INTEGRAL Applying (8) and (9) in (7), we get Representation (10) with A(w) and B(w) given by (8) and (9), respectively, is called a Fourier integral of f(x).

FOURIER INTEGRAL

EXISTENCE OF FOURIER INTEGRAL Theorem � If 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 integral exists, 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.

PROBLEMS ON FOURIER INTEGRAL EXAMPLE-1: Find the Fourier integral representation of the function Solution: The Fourier integral of the given function f(x) is

PROBLEMS ON FOURIER INTEGRAL

DIRICHLET’S DISCONTINUOUS FACTOR From Example – 1 by Fourier integral representation, we find that At 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.

DIRICHLET’S DISCONTINUOUS FACTOR The above integral is called Dirichlet’s discontinuous factor.

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 The integral as u is called the sine integral and it is denoted by Si(u)

FOURIER COSINE INTEGRAL

FOURIER COSINE INTEGRAL The Fourier integral of an even function is also known as Fourier cosine integral.

FOURIER SINE INTEGRAL

FOURIER SINE INTEGRAL The Fourier integral of an odd function is also known as Fourier sine integral.

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL Solution: The Fourier cosine integral of the given function f(x) is

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL The Fourier sine integral of the given function f(x) is

PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL Hence the Fourier sine integral of the given function is

LAPLACE INTEGRALS From Example – 2 by Fourier cosine integral representation, we find that (A)

LAPLACE INTEGRALS From Example – 2 by Fourier sine integral representation, we find that (B) The integrals (A) and (B) are called as Laplace integrals.