Higher Engineering Mathematics

662 FOURIER SERIES and (b) the sum of the Fourier series at the points of discontinuity. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (a)f(x)= 1 2 + 2 π ( cosx ...

L Fourier series 70 Fourier series for a non-periodic function over range 2π 70.1 Expansion of non-periodic functions If a funct ...

664 FOURIER SERIES = 2 π [ xsinnx n + cosnx n^2 ]π −π = 2 π [( 0 + cosnπ n^2 ) − ( 0 + cosn(−π) n^2 )] = 0 bn= 1 π ∫π −π f(x) si ...

FOURIER SERIES FOR A NON-PERIODIC FUNCTION OVER RANGE 2π 665 L an= 1 π ∫ 2 π 0 f(x) cosnxdx = 1 π [∫π 0 xcosnxdx+ ∫ 2 π π 0dx ] ...

666 FOURIER SERIES Hence π^2 8 = 1 + 1 32 + 1 52 + 1 72 +··· Problem 5. Deduce the Fourier series for the functionf(θ)=θ^2 in th ...

FOURIER SERIES FOR A NON-PERIODIC FUNCTION OVER RANGE 2π 667 L i.e. π^2 − 4 π^2 3 = 4 ( − 1 + 1 4 − 1 9 + 1 16 −··· ) − 4 π(0) − ...

668 FOURIER SERIES Sketch the waveform defined by: f(x)= ⎧ ⎪⎪ ⎪⎪ ⎨ ⎪⎪ ⎪⎪ ⎩ 1 + 2 x π , when−π<x< 0 1 − 2 x π , when 0< ...

L Fourier series 71 Even and odd functions and half-range Fourier series 71.1 Even and odd functions Even functions A functiony= ...

670 FOURIER SERIES f(x) 2 − 3 π/2 −π −π/2 0 π/2 π 3 π/2 2 π x − 2 Figure 71.1 From para. (a), a 0 = 1 π ∫π 0 f(x)dx = 1 π {∫π/ 2 ...

EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES 671 L The function is defined by: f(x)= { −2, when−π<x< 0 2, when 0 & ...

672 FOURIER SERIES i.e. π^2 − π^2 3 =− 4 ( − 1 − 1 22 − 1 32 − 1 42 − 1 52 −··· ) 2 π^2 3 = 4 ( 1 + 1 22 + 1 32 + 1 42 + 1 52 +· ...

EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES 673 L as in Section 71.2(a), i.e. f(x)=a 0 + ∑∞ n= 1 ancosnx where a 0 = 1 ...

674 FOURIER SERIES = 6 π [( −πcosnπ n + sinnπ n^2 ) −(0+0) ] =− 6 n cosnπ Whennis odd,bn= 6 n . Henceb 1 = 6 1 ,b 3 = 6 3 ,b 5 = ...

EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES 675 L or f(x)= 8 π ( 1 3 sin 2x+ 2 ( 3 )( 5 ) sin 4x + 3 ( 5 )( 7 ) sin 6x+ ...

Fourier series 72 Fourier series over any range 72.1 Expansion of a periodic function of periodL (a) A periodic function f(x) of ...

FOURIER SERIES OVER ANY RANGE 677 L v(t) 10 − 8 − 4 0 4 8 12 t (ms) Period L = 8 ms Figure 72.1 a 0 = 1 L ∫ L 2 −L 2 v(t)dt= 1 8 ...

678 FOURIER SERIES = 1 4 {∫− 1 − 2 0dx+ ∫ 1 − 1 5dx+ ∫ 2 1 0dx } = 1 4 [5x]^1 − 1 = 1 4 [(5)−(−5)]= 10 4 = 5 2 an= 2 L ∫L 2 −L 2 ...

FOURIER SERIES OVER ANY RANGE 679 L bn= 2 L ∫ L 2 −L 2 f(t) sin ( 2 πnt L ) dt = 2 L ∫L 0 tsin ( 2 πnt L ) dt = 2 3 ∫ 3 0 tsin ( ...

680 FOURIER SERIES 72.2 Half-range Fourier series for functions defined over rangeL (a) By making the substitution u= πx L (see ...

FOURIER SERIES OVER ANY RANGE 681 L Problem 5. Find the half-range Fourier sine series for the functionf(x)=xin the range 0 ≤x≤2 ...