1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

108 Chapter 1 Fourier Series and Integrals Fourier Integral Representation Theorem. Let f(x)be sectionally smooth on every finit ...

1.9 Fourier Integral 109 A(λ)=π^2 ∫∞ 0 e−xcos(λx)dx, (11) A(λ)=π^2 e −x(−cos(λx)+λsin(λx)) 1 +λ^2 ∣∣ ∣∣ ∞ 0 =π^21 +^1 λ 2. (12) ...

110 Chapter 1 Fourier Series and Integrals Sincef(x)=0 forx>π, the integral forB(λ)reduces to one over the interval 0 <x&l ...

1.9 Fourier Integral 111 f′(x)= ∫∞ 0 [ −λA(λ)sin(λx)+λB(λ)cos(λx) ] dλ. Example 5. Letf(x)=exp(−|x|)as in Example 3. Then its de ...

112 Chapter 1 Fourier Series and Integrals a.f(x)= { sin(x), −π<x<π, 0 , |x|>π; b.f(x)= { sin(x), 0 <x<π, 0 , oth ...

1.10 Complex Methods 113 v=0? Sometimes notation is compressed and, instead of the last line, we write f(x)= ∫∞ −∞ f(t)δ(t−x)dt. ...

114 Chapter 1 Fourier Series and Integrals This is the complex form of the Fourier series forf.Itiseasytoderivethe universal for ...

1.10 Complex Methods 115 Fourier integral The Fourier integral of a functionf(x)defined in the entire interval−∞< x<∞can a ...

116 Chapter 1 Fourier Series and Integrals EXERCISES 1.Use the complex form an−ibn= 1 π ∫π −π f(x)e−inxdx, n= 0 , to find the F ...

1.11 Applications of Fourier Series and Integrals 117 Find the functionf(x)whose complex Fourier coefficient function is given. ...

118 Chapter 1 Fourier Series and Integrals ̇y(t)= ∑∞ n= 1 −nAnsin(nt)+nBncos(nt), ̈y(t)= ∑∞ n= 1 −n^2 Ancos(nt)−n^2 Bnsin(nt). T ...

1.11 Applications of Fourier Series and Integrals 119 Next, suppose thatr(t)is a square-wave function with Fourier series r(t)= ...

120 Chapter 1 Fourier Series and Integrals Since the coefficients of like terms in the two series must match, we may con- clude ...

1.11 Applications of Fourier Series and Integrals 121 C. The Sampling Theorem One of the most important results of information t ...

122 Chapter 1 Fourier Series and Integrals =^1 2  ∑∞ −∞ f (nπ  ) exp (−inπω  ) , −<ω<. By utilizing Eq. (1) again, we ...

1.11 Applications of Fourier Series and Integrals 123 Figure 13 Graphs of approximation using sampling: Eq. (4) withN=100 and = ...

124 Chapter 1 Fourier Series and Integrals 5.Usethesoftwaretoapproximatethefunctionf(t)=e−t^2 by the Sampling Theorem. Try=4,N= ...

Miscellaneous Exercises 125 Chapter Review See the CD for review questions. Miscellaneous Exercises 1.Find the Fourier sine seri ...

126 Chapter 1 Fourier Series and Integrals e.odd periodic extension; f.the one corresponding tof(x)=x,−a<x<0. 8.Perform th ...

Miscellaneous Exercises 127 16.Show that the function given by the formulaf(x)=(π−x)/2, 0<x< 2 π, has the Fourier series f ...