1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

68 Chapter 1 Fourier Series and Integrals Let h(x)be an odd function defined in a symmetric interval−a<x<a. Then ∫a −a h(x ...

1.2 Arbitrary Period and Half-Range Expansions 69 (a) (b) (c) (d) Figure 4 A function is given in the interval 0<x<a(heavy ...

70 Chapter 1 Fourier Series and Integrals If the series on the right converge, theyactually represent periodic functions with pe ...

1.2 Arbitrary Period and Half-Range Expansions 71 Figure 5 Odd periodic extension (period 2) off(x)=x,0<x<1. Figure 6 Even ...

72 Chapter 1 Fourier Series and Integrals Show that the functions cos(nπx/a)and sin(nπx/a)satisfy orthogonality relations simil ...

1.3 Convergence of Fourier Series 73 12.Prove the orthogonality relations ∫a 0 sin (nπx a ) sin (mπx a ) dx= { 0 , n=m, a/ 2 , ...

74 Chapter 1 Fourier Series and Integrals (a) (b) (c) Figure 7 Three functions with different kinds of discontinuities at x=1. ( ...

1.3 Convergence of Fourier Series 75 Name Criterion Continuity f(x 0 +)=f(x 0 −)=f(x 0 ) Removable discontinuity f(x 0 +)=f(x 0 ...

76 Chapter 1 Fourier Series and Integrals The examples clarify a couple of facts about the meaning of sectional con- tinuity. Mo ...

1.3 Convergence of Fourier Series 77 sense, the value offatx=+ais unimportant. But because of the averaging features of the Four ...

78 Chapter 1 Fourier Series and Integrals a.f(x)=|x|+x, − 1 <x<1; b.f(x)=xcos(x), −π 2 <x<π 2 ; c. f(x)=xcos(x), − 1 ...

1.4 Uniform Convergence 79 The functionf(x)is periodic with period 2. Its graph for− 1 <x<1isa semicircle with radius 1 c ...

80 Chapter 1 Fourier Series and Integrals Figure 9 Partial sums of the square-wave function. Convergence isnotuniform. ...

1.4 Uniform Convergence 81 Figure 10 Partial sums of a sawtooth function. Convergence is uniform. ...

82 Chapter 1 Fourier Series and Integrals |f(x)−SN(x)|isnearlyequalto1,soconvergenceisnotuniform. (Inciden- tally, the graphs in ...

1.4 Uniform Convergence 83 Althoughf(x)is continuous and has a continuous derivative in the interval − 1 <x<1, the periodi ...

84 Chapter 1 Fourier Series and Integrals d.f(x)=sin(x)+|sin(x)|, −π<x<π; e.f(x)=x+|x|, −π<x<π; f. f(x)=x(x^2 − 1 ), ...

1.5 Operations on Fourier Series 85 1.5 Operations on Fourier Series In the course of this book we shall have to perform certain ...

86 Chapter 1 Fourier Series and Integrals ∫b a f(x)g(x)dx= ∫b a a 0 g(x)dx + ∑∞ n= 1 ∫b a ( ancos(nx)+bnsin(nx) ) g(x)dx. (3)
...

1.5 Operations on Fourier Series 87 Now, replacingbbyx,wehave x( 2 π−x) 4 = ∑∞ n= 1 1 n^2 − ∑∞ n= 1 cos(nx) n^2 , 0 ≤x≤ 2 π. (4) ...