CHAPTER 2 FOURIER SERIES


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1 CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that = sin x = sin (x + 4 ) = Therefore the function has period 2, 4, 6, etc. However, 2 is the least value and therefore is the period of f(x). Similarly cos x is a periodic function with the period 2 and tan x has period. DIRICHLET S CONDITIONS A function defined in c x c+2l can be expanded as an infinite trigonometric series of the form + provided 1. is single valued and finite in (c, c+2l) 2. is continuous or piecewise continuous with finite number of finite discontinuities in (c, c+2l). 3. has no or finite number of maxima or minima in (c, c+2l). EULER S FORMULAS If a function defined in (c, c+2l) can be expanded as the infinite trigonometric series + then [ Formulas given above for and are called Euler s formulas for Fourier coefficients]
2 DEFINITION OF FOURIER SERIES The infinite trigonometric series + is called the Fourier series of in the interval c x c+2l, provided the coefficients are given by the Euler s formulas. EVEN FUNCTION If = in (l, l) such that =, then is said to be an even function of x in (l, l). If Such that = or =, then is said to be an even function of x in (l, l). EXAMPLE y = cos x, y = are even functions. ODD FUNCTION If = in (l, l) such that = , then is said to be an odd function of x in (l, l). If Such that =  or = , then is said to be an odd function of x in (l, l). EXAMPLE y = sin x, y = x are odd functions. FOURIER SERIES OF EVEN AND ODD FUNCTIONS 1. The Fourier series of an even function in (l, l) contains only cosine terms (constant term included), i.e. the Fourier series of an even function in (l, l) is given by 2
3 = + where 2. The Fourier series of an odd function in (l, l) contains only sine terms, i.e. the Fourier series of an odd function in (l, l) is given by =, where PROBLEMS 1. Find the Fourier series of period 2l for the function = x(2l x) in (0, 2l). Deduce the sum of = Let = + in (0, 2l) (1) = using Bernoulli s formula. = 0 3
4 Using these values in (1), we have x (2l  x) = in (0, 2l)..(2) The required series can be obtained by putting x = l in the Fourier series in (2). x = l lies in (0, 2l) and is a point of continuity of the function = x(2l x). Sum the Fourier series in (2) = f(l) i.e. = l(2l  l) i.e..  = 2. Find the Fourier series of period 2 for the function = x cos x in 0 < x < 2. Let = +... (1) =0, if n 1 = 0 if n 1 4
5 if n 1 =, if n 1 = Using these values in (1), we get f(x) = 3. Find the Fourier series expansion of = sin ax in (l, l). Since is defined in a range of length 2l, we can expand in Fourier series of period 2l. Also = sin[a(x)] = sin ax =  is an odd function of x in (l, l). Hence Fourier series of will not contain cosine terms. Let =.(1) 5
6 Using these values in (1), we get 4. Find the Fourier series expansion of =. Hence obtain a series for cosec Though the range is symmetric about the origin, is neither an even function nor an odd function. Let = (1) in 6
7 Using these values in (1), we get = in i.e., [Since x=0 is a point of continuity of f(x)] i.e., i.e., 7
8 HALFRANGE FOURIER SERIES AND PARSEVAL S THEOREM (i) The half range cosine series in (0, l) is = + where (ii) The half range sine series in (0, l) is =, where (iii) The half range cosine series in (0, ) is given by = + where (iv) The half range sine series in (0, ) is given by =, where 8
9 ROOTMEAN SQUARE VALUE OF A FUNCTION Definition If a function y = is defined in (c, c+2l), then is called the root meansquare(r.m.s.) value of y in (c, c+2l) and is denoted by Thus PARSEVAL S THEOREM If y = can be expanded as a Fourier series of the form + in (c, c+2l), then the rootmean square value of y = in (c, c+2l) is given by PROOF = + in (c, c+2l)....(1) By Euler s formulas for the Fourier coefficients,.. (2).....(3) Now, by definition, = = using (1) = =, by using (2) and (3) 9
10 = EXAMPLES 1. Find the halfrange (i) cosine series and (ii) sine series for = in (0, ) (i) To get the halfrange cosine series for in (0, ), we should give an even extension for in (, 0). i.e. put = = in (, 0) Now is even in (, ). = +.(1) The Fourier halfrange cosine series of is given by in (0, ). (ii) To get the halfrange sine series of in (0, ), we should give an odd extension for in (, 0). i.e. Put =  in (, 0) =  in (, 0) Now is odd in (, ). 10
11 =.(2) Using this value in(2), we get the halfrange sine series of in (0, ). 2. Find the halfrange sine series of = sin ax in (0, l). We give an odd extension for in (l, 0). i.e. we put = sin[a(x)] = sin ax in (l, 0) is odd in (l, l) Let = 11
12 Using this values in (1), we get the halfrange sine series as 3. Find the halfrange cosine series of = a in (0, l). Deduce the sum of. Giving an odd extension for in (l, 0), is made an odd function in (l, l). Let f(x) =.. (1) Using this value in (1), we get a = Since the series whose sum is required contains constant multiples of squares of, we apply Parseval s theorem. 12
13 4. Expand =  as a Fourier series in 1 < x < 1 and using this series find the r.m.s. value of in the interval. The Fourier series of in (1, 1) is given by = +. (1).(3) 13
14 Substituting (2), (3), (4) in (1) we get We know that r.m.s. value of f(x) in (l, l) is =.(5) From (2) we get...(6) From (3) we get..(7) From (4) we get.. (8) Substituting (6), (7) and (8) in (5) we get 5. Find the Fourier series for = in Hence show that The Fourier series of in (1, 1) is given by = + 14
15 The coefficients are Parseval s theorem is i.e., = HARMONIC ANALYSIS The process of finding the Fourier series for a function given by numerical value is known as harmonic analysis. In harmonic analysis the Fourier coefficients of the function y = in (0, 2 ) are given by = 2[mean value of y in (0, 2 )] = 2[mean value of y cos nx in (0, 2 )] = 2[mean value of y sin nx in (0, 2 )] (i) Suppose the function is defined in the interval (0, 2l), then its Fourier series is, = + and now, = 2[mean value of y in (0, 2l)] = 15
16 = (ii) If the half range Fourier sine series of in (0, l) is, =, then = (iii) If the half range Fourier sine series of in (0, ) is, =, then = (iv) If the half range Fourier cosine series of in (0, l) is, = +, then = 2[mean value of y in (0, l)] = (v) If the half range Fourier cosine series of in (0, ) is, = +, then = 2[mean value of y in (0, )] =. EXAMPLES 1. The following table gives the variations of a periodic function over a period T. x Show that = , where 16
17 Here the last value is a mere repetition of the first therefore we omit that value and consider the remaining 6 values. n = 6. Given....(1) when x takes the values of 0,,,,, takes the values 0,,,,,. (By using (1)) Let the Fourier series be of the form (2) where n = 6 y cos sin y cos y sin , Substituting these values of in (2), we get = cos sin 17
18 2. Find the Fourier series upto the third harmonic for the function y = defined in (0, ) from the table x We can express the given data in a half range Fourier sine series......(1) Now x y = f(0) sin x sin 2x sin 3x y sin x y sin 2x y sin 3x Substituting these values in (1), we get = 1.4 sin x sin 2x sin 3x 3. Compute the first two harmonics of the Fourier series for f(x) from the following data x
19 Here the length of the interval is Fourier sine series i.e., we can express the given data in a half range (1) x y sin x sin 2x Now = sin x sin 2x 4. Find the Fourier series as far as the second harmonic to represent the function given in the following data. x Here the length of the interval is 6 (not 2 ) i.e., 2l = 6 or l = 3 The Fourier series is..(1) y
20 Substituting these values of in (1), we get COMPLEX FORM OF FOURIER SERIES The equation of the form is called the complex form or exponential form of the Fourier series of in (c, c+2l). The coefficient is given by When l =, the complex form of Fourier series of in (c, c+2 ) takes the form where 20
21 PROBLEMS 1. Find the complex form of the Fourier series of = in (0, 2). Since 2l = 2 or l = 1, the complex form of the Fourier series is Using this value in (1), we get 2. Find the complex form of the Fourier series of = sin x in (0, ). Here 2l = or l =. The complex form of Fourier series is..(1) 21
22 Using this value in (1), we get in (0, ) 3. Find the complex form of the Fourier series of = in (l, l). Let the complex form of the Fourier series be Using this value in (1), we have 22
23 in (l, l) 4. Find the complex form of the Fourier series of = cos ax in (, ), where a is neither zero nor an integer. Here 2l = 2 or l =. The complex form of Fourier series is.(1) Using this value in (1), we get in (, ). UNIT 2 PART A 1. Determine the value of in the Fourier series expansion of Ans: is an odd function. 2. Find the root mean square value of in the interval. Ans: RMS Vale of in is 23
24 3. Find the coefficient of in the Fourier cosine series of the function in the interval Ans: Here Fourier cosine series is = +, where 4. If and for all x, find the sum of the Fourier series of at. Ans: Here is a point of discontinuity. The sum of the Fourier series is equal to the average of right hand and left hand limit of the given function at. i.e., 5. Find in the expansion of as a Fourier series in. Ans: = 0 Since is an even function in. 6. If is an odd function defined in (l, l) what are the values of 24
25 Ans: = 0 since is an odd function. 7. Find the Fourier constants for in. Ans: = 0 Since is an even function in. 8. State Parseval s identity for the halfrange cosine expansion of in (0, 1). Ans: where 9. Find the constant term in the Fourier series expansion of in. Ans: = 0 since is an odd function in. 10. State Dirichlet s conditions for Fourier series. Ans: (i) is defined and single valued except possibly at a finite number of points in. (ii) is periodic with period 2. (iii) and are piecewise continuous in. Then the Fourier series of converges to (a) (b) if x is a point of continuity if x is a point of discontinuity. 11. What you mean by Harmonic Analysis? Ans: 25
26 The process of finding the Fourier series for a function given by numerical value is known as harmonic analysis. In harmonic analysis the Fourier coefficients of the function y = in (0, 2 ) are given by = 2[mean value of y in (0, 2 )] = 2[mean value of y cos nx in (0, 2 )] = 2[mean value of y sin nx in (0, 2 )] 12. In the Fourier expansion of in. Find the value of, the coefficient of sin nx. Ans: Since is an even function the value of = What is the constant term and the coefficient of in the Fourier expansion of in (7, 7)? Ans: Given The given function is an odd function. Hence are zero. 14. State Parseval s identity for full range expansion of as Fourier series in (0, 2l). Ans: = where 26
27 15. Find a Fourier sine series for the function = 1; 0 < x <. Ans: The Fourier sine series of.(1) 16. If the Fourier series for the function is Ans: Deduce that Putting we get 27
28 17. Define Root mean square value of a function? Ans: If a function y = is defined in (c, c+2l), then is called the root meansquare(r.m.s.) value of y in (c, c+2l) and is denoted by Thus 18. If is expressed as a Fourier series in the interval (2, 2), to which value this series converges at x = 2. Ans: Since x = 2 is a point of continuity, the Fourier series converges to the arithmetic mean of at x = 2 and x = 2 i.e., 19. If the Fourier series corresponding to in the interval is without finding the values of find the value of Ans: By using Parseval s identity, 20. Find the constant term in the Fourier series corresponding to expressed in the interval. Ans: Given Now 28
29 PART B 1. (i) Express as a Fourier series in (ii) Show that for 0 < x <l,. Using root mean square value of x, deduce the value of 2. (i) Find the Fourier series of periodicity 3 for in 0 < x < 3. (ii) Find the Fourier series expansion of period 2 for the function which is defined in by means of the table of values given below. Find the series upto the third harmonic. x (i) Find the Fourier series of periodicity 2 for for 0 < x < 2. (ii) Show that for 0 < x <l,. Deduce that 4. (i) Find the Fourier series for. Hence deduce the sum to infinity of the series (ii) Find the complex form of Fourier series of in the form and hence prove that 5. Obtain the half range cosine series for in 6. Find the Fourier series for in the interval. 7. (i) Expanding as a sine series in show that (ii) Find the Fourier series as far as the second harmonic to represent the function given in the following data. 29
30 x Obtain the Fourier series for of period 2l and defined as follows Hence deduce that 9. Obtain the half range cosine series for in 10. (i) Find the Fourier series of (ii) Obtain the sine series for the function 11. (i) Find the Fourier series for the function and for all x. (ii) Determine the Fourier series for the function 12. Obtain the Fourier series for in. Deduce that 13. Obtain the constant term and the first harmonic in the Fourier series expansion for where is given in the following table. x
31 14. (i) Express as a Fourier series in (ii) Obtain the half range cosine series for in the interval 0 < x < Find the half range sine series of in 16. (i) Find the Fourier series expansion of = (ii) Find the halfrange sine series of = sin ax in (0, l). 17. Expand =  as a Fourier series in 1 < x < 1 and using this series find the r.m.s. value of in the interval. 18. The following table gives the variations of a periodic function over a period T. x Show that = , where 19. Find the Fourier series upto the third harmonic for the function y = defined in (0, ) from the table x (i) Find the halfrange (i) cosine series and (ii) sine series for = in (0, ) (ii) Find the complex form of the Fourier series of = cos ax in (, ). 31
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