Page 1 Free coaching of B.Sc (h) maths & JAM For more 8130648819 FOURIER SERIES Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier coefficient, Fourier theorem, discussion of the theorem and its corollary. Reference: John P. Dâ€™ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. Fourier Series A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x Result 1. ? f(x)dx = ? f(x)dx 2. ? f(x)dx = ? f(x)dx 3. ? f(x)dx =? f( x)dx 4. ? cosmxcosnx dx ={ 0 2 for m n for m n 0 for m n 0 5. ? sinmxsinnx = { 0 0 for m n for m n 0 for m n 0 6. ? cosmxsinnx =0 for all m, n +ve integer 7 ? cosnx dx = 2 0 2 if n 0 if n 0 8 ? sinnx dx = 0 When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat conduction, he needed to express a function f as an infinite series of sine and cosine function (called a trigonometric series) f(x) a ?(a cosnx b sinnx) .(i) We start by assuming [assume â€“ to think or accept that sth is true but without having proof of it] that the trigonometric series converges uniformly on , , - to the continuous sum function f(x): we integrate both sides of equation (i),using term by term integration valid due to uniform convergence,we obtain ? f(x)dx ? a dx ? ?(a cosnx b sinnx) dx a .2 ? a ? cosnx dx ? b ? sinnxdx a .2 0 0 a 1 2 ? f(x)dx .(ii) To determine a n for n 1 ,we multiply both sides of equation f(x) a ?(a cosmx b sinmx) .(iii) bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not destroyed,and so term by term of integration is justified) and intergrate term by term from to justified (in doing sth) having a good reason for doing sth ? f(x)cosnxdx a ? cosnxdx ? a cosmxcosnxdx ? b ? sinmx cosnxdx 0 a 0 Page 2 Free coaching of B.Sc (h) maths & JAM For more 8130648819 FOURIER SERIES Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier coefficient, Fourier theorem, discussion of the theorem and its corollary. Reference: John P. Dâ€™ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. Fourier Series A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x Result 1. ? f(x)dx = ? f(x)dx 2. ? f(x)dx = ? f(x)dx 3. ? f(x)dx =? f( x)dx 4. ? cosmxcosnx dx ={ 0 2 for m n for m n 0 for m n 0 5. ? sinmxsinnx = { 0 0 for m n for m n 0 for m n 0 6. ? cosmxsinnx =0 for all m, n +ve integer 7 ? cosnx dx = 2 0 2 if n 0 if n 0 8 ? sinnx dx = 0 When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat conduction, he needed to express a function f as an infinite series of sine and cosine function (called a trigonometric series) f(x) a ?(a cosnx b sinnx) .(i) We start by assuming [assume â€“ to think or accept that sth is true but without having proof of it] that the trigonometric series converges uniformly on , , - to the continuous sum function f(x): we integrate both sides of equation (i),using term by term integration valid due to uniform convergence,we obtain ? f(x)dx ? a dx ? ?(a cosnx b sinnx) dx a .2 ? a ? cosnx dx ? b ? sinnxdx a .2 0 0 a 1 2 ? f(x)dx .(ii) To determine a n for n 1 ,we multiply both sides of equation f(x) a ?(a cosmx b sinmx) .(iii) bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not destroyed,and so term by term of integration is justified) and intergrate term by term from to justified (in doing sth) having a good reason for doing sth ? f(x)cosnxdx a ? cosnxdx ? a cosmxcosnxdx ? b ? sinmx cosnxdx 0 a 0 Free coaching of B.Sc (h) maths & JAM For more 8130648819 a 1 ? f(x)cosnxdx, n 1 ,2,3, . eq (iv) similary,if we multiply both sides of the eq (iii) bysinnx and integrate from to ,we get b 1 ? f(x)sinnx dx,n 1 ,2,3, eq (v) Definition A trigonometric series of the form a ?(a cosnx b sinnx) ..(vi) Is called Fouries series of a function f periodic with 2 if the Fourier coefficients a 0, a n, b n are determined by (ii) , (iii) & (v) note Since,|a cosnx b sinnx| |a | |b |, n 1 by ei erstrass s,M test,it follows that,if the series ?(|a | |b |) converges, then the trigonometric series (vi)converges absolutely and uniformly in , , - and it is the Fourier series of a continuous 2 periodic fuction f(x).Moreover ?(a b ) and so lim a 0 , lim b 0 . Alternative way of defining Fourier Series A series is +? (a cosnx b sinnx) is called a trigonometric series and the constant a 0, a n &b n are called coefficient. If this series converges its sum is periodic and its period is 2 . (ii) let f be bounded and integrable on , , - then the series +? (a cosnx b sinnx) where a n = ? f(x)cosnxdx ? n 0, 1, 2 .. and b n = ? f(x)sinnxdx ? n 0, 1, 2 .. (1) is called the Fourier series of the function on , , ] and the coefficient is given by the equation (1) are called Fourier coefficient of the function. Imp.thm. Let f R R be such that (i) f is bounded and integrable on , , ] (ii) The function f is piece wise monotonic on , , ] (iii) f is periodic with period 2 i.e. f(x+2 ) = f(x) ? x ? R then a ?(a cosnx b sinnx) { f(x),for every point x of continuity in , , - 1 2 ,f(x ) f(x )-,for x 1 2 ,f( ) f( )-,for x ± Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left- hand and right-hand limits of the function f at points x in - , , of discontinuity, of the function f at points x in - , , of discontinuity,and using periodicity of f,i.e.f(x 2 ) f(x), 1 2 ,f( ) f( )-at the end points ± Theorem: Let f : R R be such that (i) f is bounded and integrable on , , ] (ii) f is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? < ) then +? a = ( ) ( ) M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by f(x) = 2 x x x 0 0 x what is the sum of the fourier series function x 0 and x . Deduce that = 1+ (1) Check SV notes pg 11 Sol n : f(x) is bounded and integrable on , , ] Page 3 Free coaching of B.Sc (h) maths & JAM For more 8130648819 FOURIER SERIES Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier coefficient, Fourier theorem, discussion of the theorem and its corollary. Reference: John P. Dâ€™ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. Fourier Series A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x Result 1. ? f(x)dx = ? f(x)dx 2. ? f(x)dx = ? f(x)dx 3. ? f(x)dx =? f( x)dx 4. ? cosmxcosnx dx ={ 0 2 for m n for m n 0 for m n 0 5. ? sinmxsinnx = { 0 0 for m n for m n 0 for m n 0 6. ? cosmxsinnx =0 for all m, n +ve integer 7 ? cosnx dx = 2 0 2 if n 0 if n 0 8 ? sinnx dx = 0 When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat conduction, he needed to express a function f as an infinite series of sine and cosine function (called a trigonometric series) f(x) a ?(a cosnx b sinnx) .(i) We start by assuming [assume â€“ to think or accept that sth is true but without having proof of it] that the trigonometric series converges uniformly on , , - to the continuous sum function f(x): we integrate both sides of equation (i),using term by term integration valid due to uniform convergence,we obtain ? f(x)dx ? a dx ? ?(a cosnx b sinnx) dx a .2 ? a ? cosnx dx ? b ? sinnxdx a .2 0 0 a 1 2 ? f(x)dx .(ii) To determine a n for n 1 ,we multiply both sides of equation f(x) a ?(a cosmx b sinmx) .(iii) bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not destroyed,and so term by term of integration is justified) and intergrate term by term from to justified (in doing sth) having a good reason for doing sth ? f(x)cosnxdx a ? cosnxdx ? a cosmxcosnxdx ? b ? sinmx cosnxdx 0 a 0 Free coaching of B.Sc (h) maths & JAM For more 8130648819 a 1 ? f(x)cosnxdx, n 1 ,2,3, . eq (iv) similary,if we multiply both sides of the eq (iii) bysinnx and integrate from to ,we get b 1 ? f(x)sinnx dx,n 1 ,2,3, eq (v) Definition A trigonometric series of the form a ?(a cosnx b sinnx) ..(vi) Is called Fouries series of a function f periodic with 2 if the Fourier coefficients a 0, a n, b n are determined by (ii) , (iii) & (v) note Since,|a cosnx b sinnx| |a | |b |, n 1 by ei erstrass s,M test,it follows that,if the series ?(|a | |b |) converges, then the trigonometric series (vi)converges absolutely and uniformly in , , - and it is the Fourier series of a continuous 2 periodic fuction f(x).Moreover ?(a b ) and so lim a 0 , lim b 0 . Alternative way of defining Fourier Series A series is +? (a cosnx b sinnx) is called a trigonometric series and the constant a 0, a n &b n are called coefficient. If this series converges its sum is periodic and its period is 2 . (ii) let f be bounded and integrable on , , - then the series +? (a cosnx b sinnx) where a n = ? f(x)cosnxdx ? n 0, 1, 2 .. and b n = ? f(x)sinnxdx ? n 0, 1, 2 .. (1) is called the Fourier series of the function on , , ] and the coefficient is given by the equation (1) are called Fourier coefficient of the function. Imp.thm. Let f R R be such that (i) f is bounded and integrable on , , ] (ii) The function f is piece wise monotonic on , , ] (iii) f is periodic with period 2 i.e. f(x+2 ) = f(x) ? x ? R then a ?(a cosnx b sinnx) { f(x),for every point x of continuity in , , - 1 2 ,f(x ) f(x )-,for x 1 2 ,f( ) f( )-,for x ± Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left- hand and right-hand limits of the function f at points x in - , , of discontinuity, of the function f at points x in - , , of discontinuity,and using periodicity of f,i.e.f(x 2 ) f(x), 1 2 ,f( ) f( )-at the end points ± Theorem: Let f : R R be such that (i) f is bounded and integrable on , , ] (ii) f is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? < ) then +? a = ( ) ( ) M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by f(x) = 2 x x x 0 0 x what is the sum of the fourier series function x 0 and x . Deduce that = 1+ (1) Check SV notes pg 11 Sol n : f(x) is bounded and integrable on , , ] Free coaching of B.Sc (h) maths & JAM For more 8130648819 a ? f(x)cosnxdx = 0? f(x)cosnxdx ? f(x)cosnxdx 1 = 0? (x )cosnxdx ? ( x)cosnxdx 1= I 1+I 2 (i) Consider, I 1 = ? (x )cos nx dx Let u x ? dx du I 1 =? ( u )cosn( u) ( du) = ? (u )cos nu du ? (u )cos nu du ? ( x) cos nx dx a n = ? ( x x) cosnx dx (ii) ? xcosnx dx 0.x / ? 1 . dx 1 0 (cosnx) 1 ,cosn cos0 - ,( 1 ) 1 - a n = ,1 ( 1 ) - Put n = 0 in equation (ii) a = ? x dx . (x ) Similarly, b n = *1 ( 1 ) + b n = ? f(x)sinnx dx = ? f(x)sinnx dx +? f(x)sin nx dx = ? (x )sin nx dx +? ( x)sinnx dx Consider, I 1 = ? (x )sinnx dx Let u x ? du dx I 1 = ? ( u )sinn( u)( du) ? (u )sin nudu =? (u )sin nudu =? (x )sinnxdx by change of variable b n = [? (x x)sinnxdx ] = ? 2 sinnxdx = 20 1 2 0 ( ) 1 = ,1 ( 1 ) - b n = ,1 ( 1 ) - ? a n = ,0 , ,0 , ,0 , ? b n = 4 1 ,0 , ,0 , ,0 , the fourier series of the function f(x) f(x) +02 3 2 4 31 (iii) The sum of the fourier series at x 0 is ( ) ( ) = =0 ,? f(0 )= lim f(x)= lim (x ?? )= lim (0 h ?? ) ?? The sum of the fourier series at x = ± is ( ) ( ) lim f(x)= lim (?? x)= lim ?? (?? h ) = 0 0? lim f(x) lim (x ) lim ( h ) 2 1 Sum of the fourier series at x 0 is f(0) put x 0 in equation (iii) Page 4 Free coaching of B.Sc (h) maths & JAM For more 8130648819 FOURIER SERIES Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier coefficient, Fourier theorem, discussion of the theorem and its corollary. Reference: John P. Dâ€™ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. Fourier Series A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x Result 1. ? f(x)dx = ? f(x)dx 2. ? f(x)dx = ? f(x)dx 3. ? f(x)dx =? f( x)dx 4. ? cosmxcosnx dx ={ 0 2 for m n for m n 0 for m n 0 5. ? sinmxsinnx = { 0 0 for m n for m n 0 for m n 0 6. ? cosmxsinnx =0 for all m, n +ve integer 7 ? cosnx dx = 2 0 2 if n 0 if n 0 8 ? sinnx dx = 0 When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat conduction, he needed to express a function f as an infinite series of sine and cosine function (called a trigonometric series) f(x) a ?(a cosnx b sinnx) .(i) We start by assuming [assume â€“ to think or accept that sth is true but without having proof of it] that the trigonometric series converges uniformly on , , - to the continuous sum function f(x): we integrate both sides of equation (i),using term by term integration valid due to uniform convergence,we obtain ? f(x)dx ? a dx ? ?(a cosnx b sinnx) dx a .2 ? a ? cosnx dx ? b ? sinnxdx a .2 0 0 a 1 2 ? f(x)dx .(ii) To determine a n for n 1 ,we multiply both sides of equation f(x) a ?(a cosmx b sinmx) .(iii) bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not destroyed,and so term by term of integration is justified) and intergrate term by term from to justified (in doing sth) having a good reason for doing sth ? f(x)cosnxdx a ? cosnxdx ? a cosmxcosnxdx ? b ? sinmx cosnxdx 0 a 0 Free coaching of B.Sc (h) maths & JAM For more 8130648819 a 1 ? f(x)cosnxdx, n 1 ,2,3, . eq (iv) similary,if we multiply both sides of the eq (iii) bysinnx and integrate from to ,we get b 1 ? f(x)sinnx dx,n 1 ,2,3, eq (v) Definition A trigonometric series of the form a ?(a cosnx b sinnx) ..(vi) Is called Fouries series of a function f periodic with 2 if the Fourier coefficients a 0, a n, b n are determined by (ii) , (iii) & (v) note Since,|a cosnx b sinnx| |a | |b |, n 1 by ei erstrass s,M test,it follows that,if the series ?(|a | |b |) converges, then the trigonometric series (vi)converges absolutely and uniformly in , , - and it is the Fourier series of a continuous 2 periodic fuction f(x).Moreover ?(a b ) and so lim a 0 , lim b 0 . Alternative way of defining Fourier Series A series is +? (a cosnx b sinnx) is called a trigonometric series and the constant a 0, a n &b n are called coefficient. If this series converges its sum is periodic and its period is 2 . (ii) let f be bounded and integrable on , , - then the series +? (a cosnx b sinnx) where a n = ? f(x)cosnxdx ? n 0, 1, 2 .. and b n = ? f(x)sinnxdx ? n 0, 1, 2 .. (1) is called the Fourier series of the function on , , ] and the coefficient is given by the equation (1) are called Fourier coefficient of the function. Imp.thm. Let f R R be such that (i) f is bounded and integrable on , , ] (ii) The function f is piece wise monotonic on , , ] (iii) f is periodic with period 2 i.e. f(x+2 ) = f(x) ? x ? R then a ?(a cosnx b sinnx) { f(x),for every point x of continuity in , , - 1 2 ,f(x ) f(x )-,for x 1 2 ,f( ) f( )-,for x ± Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left- hand and right-hand limits of the function f at points x in - , , of discontinuity, of the function f at points x in - , , of discontinuity,and using periodicity of f,i.e.f(x 2 ) f(x), 1 2 ,f( ) f( )-at the end points ± Theorem: Let f : R R be such that (i) f is bounded and integrable on , , ] (ii) f is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? < ) then +? a = ( ) ( ) M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by f(x) = 2 x x x 0 0 x what is the sum of the fourier series function x 0 and x . Deduce that = 1+ (1) Check SV notes pg 11 Sol n : f(x) is bounded and integrable on , , ] Free coaching of B.Sc (h) maths & JAM For more 8130648819 a ? f(x)cosnxdx = 0? f(x)cosnxdx ? f(x)cosnxdx 1 = 0? (x )cosnxdx ? ( x)cosnxdx 1= I 1+I 2 (i) Consider, I 1 = ? (x )cos nx dx Let u x ? dx du I 1 =? ( u )cosn( u) ( du) = ? (u )cos nu du ? (u )cos nu du ? ( x) cos nx dx a n = ? ( x x) cosnx dx (ii) ? xcosnx dx 0.x / ? 1 . dx 1 0 (cosnx) 1 ,cosn cos0 - ,( 1 ) 1 - a n = ,1 ( 1 ) - Put n = 0 in equation (ii) a = ? x dx . (x ) Similarly, b n = *1 ( 1 ) + b n = ? f(x)sinnx dx = ? f(x)sinnx dx +? f(x)sin nx dx = ? (x )sin nx dx +? ( x)sinnx dx Consider, I 1 = ? (x )sinnx dx Let u x ? du dx I 1 = ? ( u )sinn( u)( du) ? (u )sin nudu =? (u )sin nudu =? (x )sinnxdx by change of variable b n = [? (x x)sinnxdx ] = ? 2 sinnxdx = 20 1 2 0 ( ) 1 = ,1 ( 1 ) - b n = ,1 ( 1 ) - ? a n = ,0 , ,0 , ,0 , ? b n = 4 1 ,0 , ,0 , ,0 , the fourier series of the function f(x) f(x) +02 3 2 4 31 (iii) The sum of the fourier series at x 0 is ( ) ( ) = =0 ,? f(0 )= lim f(x)= lim (x ?? )= lim (0 h ?? ) ?? The sum of the fourier series at x = ± is ( ) ( ) lim f(x)= lim (?? x)= lim ?? (?? h ) = 0 0? lim f(x) lim (x ) lim ( h ) 2 1 Sum of the fourier series at x 0 is f(0) put x 0 in equation (iii) Free coaching of B.Sc (h) maths & JAM For more 8130648819 f(0) = 01 1+0 0 = 01 1+0 . = 1 = 1 Que. If f(x)=2 cosx cosx x 0 0 x P/T the fourier series of the function f(x) is 0 . sin2x . sin4x . sin6x 1 Sol n Since the function f(x) is bounded and integrable on , ?? , ?? ] a ? f(x)cosnxdx = 0? f(x) cosnxdx ? f(x)cosnxdx 1 = I 1+ I 2 Consider, I 1 = ? f(x) cosnxdx u x ? dx du I 1 =? cos( u)cosn( u)( du) ? cosucosnudu =? cosucosnudu =? cosxcosnxdx a n ? cosxcosnxdx + ? cosxcosnxdx = 0 b n = ? f(x)sinnxdx = ? cosxsinnxdx + ? cosxsinnxdx Let I 1= ? cosxsinnxdx ? cos( u)sinn( u)( du) Put x u , dx du = ? cosusinnu du = ? cosxsinnxdx b n = ? cosxsinnxdx = ? ,sin(n 1 )x sin(n 1 )x-dx = 0 ( ) ( ) 1 0 ( ) ( ) 1 0 *cos(n 1 ) cos0 + *cos(n 1 ) cos0 +1 0 *( 1 ) 1 + *( 1 ) 1 +1 ,( 1 ) 1 -0 1 *? ( 1 ) ( 1 ) ,( 1 ) 1 - ( ) = . ,1 ( 1 ) - = . ,1 ( 1 ) ( 1 )- b n = . ,1 ( 1 ) - b n =.0 , , ,0 , , ,0 ./ Since, a 0=0 and a n=0 the fourier series is f(x) = ? b sin nx Page 5 Free coaching of B.Sc (h) maths & JAM For more 8130648819 FOURIER SERIES Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier coefficient, Fourier theorem, discussion of the theorem and its corollary. Reference: John P. Dâ€™ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. Fourier Series A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x Result 1. ? f(x)dx = ? f(x)dx 2. ? f(x)dx = ? f(x)dx 3. ? f(x)dx =? f( x)dx 4. ? cosmxcosnx dx ={ 0 2 for m n for m n 0 for m n 0 5. ? sinmxsinnx = { 0 0 for m n for m n 0 for m n 0 6. ? cosmxsinnx =0 for all m, n +ve integer 7 ? cosnx dx = 2 0 2 if n 0 if n 0 8 ? sinnx dx = 0 When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat conduction, he needed to express a function f as an infinite series of sine and cosine function (called a trigonometric series) f(x) a ?(a cosnx b sinnx) .(i) We start by assuming [assume â€“ to think or accept that sth is true but without having proof of it] that the trigonometric series converges uniformly on , , - to the continuous sum function f(x): we integrate both sides of equation (i),using term by term integration valid due to uniform convergence,we obtain ? f(x)dx ? a dx ? ?(a cosnx b sinnx) dx a .2 ? a ? cosnx dx ? b ? sinnxdx a .2 0 0 a 1 2 ? f(x)dx .(ii) To determine a n for n 1 ,we multiply both sides of equation f(x) a ?(a cosmx b sinmx) .(iii) bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not destroyed,and so term by term of integration is justified) and intergrate term by term from to justified (in doing sth) having a good reason for doing sth ? f(x)cosnxdx a ? cosnxdx ? a cosmxcosnxdx ? b ? sinmx cosnxdx 0 a 0 Free coaching of B.Sc (h) maths & JAM For more 8130648819 a 1 ? f(x)cosnxdx, n 1 ,2,3, . eq (iv) similary,if we multiply both sides of the eq (iii) bysinnx and integrate from to ,we get b 1 ? f(x)sinnx dx,n 1 ,2,3, eq (v) Definition A trigonometric series of the form a ?(a cosnx b sinnx) ..(vi) Is called Fouries series of a function f periodic with 2 if the Fourier coefficients a 0, a n, b n are determined by (ii) , (iii) & (v) note Since,|a cosnx b sinnx| |a | |b |, n 1 by ei erstrass s,M test,it follows that,if the series ?(|a | |b |) converges, then the trigonometric series (vi)converges absolutely and uniformly in , , - and it is the Fourier series of a continuous 2 periodic fuction f(x).Moreover ?(a b ) and so lim a 0 , lim b 0 . Alternative way of defining Fourier Series A series is +? (a cosnx b sinnx) is called a trigonometric series and the constant a 0, a n &b n are called coefficient. If this series converges its sum is periodic and its period is 2 . (ii) let f be bounded and integrable on , , - then the series +? (a cosnx b sinnx) where a n = ? f(x)cosnxdx ? n 0, 1, 2 .. and b n = ? f(x)sinnxdx ? n 0, 1, 2 .. (1) is called the Fourier series of the function on , , ] and the coefficient is given by the equation (1) are called Fourier coefficient of the function. Imp.thm. Let f R R be such that (i) f is bounded and integrable on , , ] (ii) The function f is piece wise monotonic on , , ] (iii) f is periodic with period 2 i.e. f(x+2 ) = f(x) ? x ? R then a ?(a cosnx b sinnx) { f(x),for every point x of continuity in , , - 1 2 ,f(x ) f(x )-,for x 1 2 ,f( ) f( )-,for x ± Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left- hand and right-hand limits of the function f at points x in - , , of discontinuity, of the function f at points x in - , , of discontinuity,and using periodicity of f,i.e.f(x 2 ) f(x), 1 2 ,f( ) f( )-at the end points ± Theorem: Let f : R R be such that (i) f is bounded and integrable on , , ] (ii) f is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? < ) then +? a = ( ) ( ) M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by f(x) = 2 x x x 0 0 x what is the sum of the fourier series function x 0 and x . Deduce that = 1+ (1) Check SV notes pg 11 Sol n : f(x) is bounded and integrable on , , ] Free coaching of B.Sc (h) maths & JAM For more 8130648819 a ? f(x)cosnxdx = 0? f(x)cosnxdx ? f(x)cosnxdx 1 = 0? (x )cosnxdx ? ( x)cosnxdx 1= I 1+I 2 (i) Consider, I 1 = ? (x )cos nx dx Let u x ? dx du I 1 =? ( u )cosn( u) ( du) = ? (u )cos nu du ? (u )cos nu du ? ( x) cos nx dx a n = ? ( x x) cosnx dx (ii) ? xcosnx dx 0.x / ? 1 . dx 1 0 (cosnx) 1 ,cosn cos0 - ,( 1 ) 1 - a n = ,1 ( 1 ) - Put n = 0 in equation (ii) a = ? x dx . (x ) Similarly, b n = *1 ( 1 ) + b n = ? f(x)sinnx dx = ? f(x)sinnx dx +? f(x)sin nx dx = ? (x )sin nx dx +? ( x)sinnx dx Consider, I 1 = ? (x )sinnx dx Let u x ? du dx I 1 = ? ( u )sinn( u)( du) ? (u )sin nudu =? (u )sin nudu =? (x )sinnxdx by change of variable b n = [? (x x)sinnxdx ] = ? 2 sinnxdx = 20 1 2 0 ( ) 1 = ,1 ( 1 ) - b n = ,1 ( 1 ) - ? a n = ,0 , ,0 , ,0 , ? b n = 4 1 ,0 , ,0 , ,0 , the fourier series of the function f(x) f(x) +02 3 2 4 31 (iii) The sum of the fourier series at x 0 is ( ) ( ) = =0 ,? f(0 )= lim f(x)= lim (x ?? )= lim (0 h ?? ) ?? The sum of the fourier series at x = ± is ( ) ( ) lim f(x)= lim (?? x)= lim ?? (?? h ) = 0 0? lim f(x) lim (x ) lim ( h ) 2 1 Sum of the fourier series at x 0 is f(0) put x 0 in equation (iii) Free coaching of B.Sc (h) maths & JAM For more 8130648819 f(0) = 01 1+0 0 = 01 1+0 . = 1 = 1 Que. If f(x)=2 cosx cosx x 0 0 x P/T the fourier series of the function f(x) is 0 . sin2x . sin4x . sin6x 1 Sol n Since the function f(x) is bounded and integrable on , ?? , ?? ] a ? f(x)cosnxdx = 0? f(x) cosnxdx ? f(x)cosnxdx 1 = I 1+ I 2 Consider, I 1 = ? f(x) cosnxdx u x ? dx du I 1 =? cos( u)cosn( u)( du) ? cosucosnudu =? cosucosnudu =? cosxcosnxdx a n ? cosxcosnxdx + ? cosxcosnxdx = 0 b n = ? f(x)sinnxdx = ? cosxsinnxdx + ? cosxsinnxdx Let I 1= ? cosxsinnxdx ? cos( u)sinn( u)( du) Put x u , dx du = ? cosusinnu du = ? cosxsinnxdx b n = ? cosxsinnxdx = ? ,sin(n 1 )x sin(n 1 )x-dx = 0 ( ) ( ) 1 0 ( ) ( ) 1 0 *cos(n 1 ) cos0 + *cos(n 1 ) cos0 +1 0 *( 1 ) 1 + *( 1 ) 1 +1 ,( 1 ) 1 -0 1 *? ( 1 ) ( 1 ) ,( 1 ) 1 - ( ) = . ,1 ( 1 ) - = . ,1 ( 1 ) ( 1 )- b n = . ,1 ( 1 ) - b n =.0 , , ,0 , , ,0 ./ Since, a 0=0 and a n=0 the fourier series is f(x) = ? b sin nx Free coaching of B.Sc (h) maths & JAM For more 8130648819 = 0 . sin2x . sin4x 1 Que. Show that Cos kx = 0 1 ?? x ?? k being non integr al. Deduce that ? ?? cot kx = 2k ? ? ? ( 1 ) 0 1 Sol n f(x)= cos kx ?? x ?? a n = ? f(x)cosnxdx = ? coskxcosnxdx = ? coskxcosnxdx = ? cos(k n) cos(k n)xdx Now, sin(k+n)?? = sin k?? cos n?? +cos k?? sin n?? =( 1 ) sin k?? + 0 =( 1 ) sin k?? Similary, sin(k n)?? = ( 1 ) sin k?? a n = ( ) 0 1 = ( ) 0 ( )( ) 1 a 0 = . = . b n = ? f(x)sinnx dx = ? coskxsinnx dx 0 *? of odd function + b n = 0 The fourier series of the function f(x) is a 2 ? a cosnx sink . 1 k ? ( 1 ) sink . 2k k n cosnx = + ? ( ) cosnx Cos kx= 0 2k ? ( ) cosnx 1 Since , the function f(x) is continuous on , ?? , ?? ] fourier series at x f(x) ? x? , ?? , ?? ]. Deduction (i)In particular the fourier series at ?? =f(?? ) Cos k?? = 0 2k ? ( ) cosn 1 ?? cot k?? = 2k ? ( ) ( ) ?? cot k?? = 2k ? (ii)In particular the fourier series at 0=f(0) Cos 0 = 0 2k ? ( ) 1 ? ( ) = ? ( 1 ) 0 1 =2 ? ( 1 ) . 3 ? ( ) =2? ( 1 ) 3 ? ( 1 ) ( ) =? ( 1 ) 0 ( ) 1 Que letRead More

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