1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

88 Chapter 1 Fourier Series and Integrals uous, it is certain that the differentiated series off(x)will fail to converge, at som ...

1.5 Operations on Fourier Series 89 in whichαis a positive parameter. For this function we havea 0 =0,an=e−nα, bn=0. By the inte ...

90 Chapter 1 Fourier Series and Integrals Use the series that follows, together with integration or differentiation, to find a ...

1.6 Mean Error and Convergence in Mean 91 is a finite number. Let f(x)∼a 0 + ∑∞ n= 1 ancos (nπx a ) +bnsin (nπx a ) and letg(x)h ...

92 Chapter 1 Fourier Series and Integrals already have an expression for the middle integral. The last one can be found by repla ...

1.6 Mean Error and Convergence in Mean 93 This inequality is valid for anyNand therefore is also valid in the limit asN tends to ...

94 Chapter 1 Fourier Series and Integrals 3.an= 1 a ∫a −a f(x)cos (nπx a ) dx→0asn→∞; bn=^1 a ∫a −a f(x)sin ( nπx a ) dx→0asn→∞. ...

1.7 Proof of Convergence 95 Use the fact that ∫∞ 0 sin(t) t dt= π 2. 1.7 Proof of Convergence In this section we prove the Fouri ...

96 Chapter 1 Fourier Series and Integrals Proof: Let the pointxbe chosen; it is to remain fixed. To begin with, we assume thatfi ...

1.7 Proof of Convergence 97 = 1 π ∫π −π f(z) ( 1 2 + ∑N n= 1 cos(nz)cos(nx)+sin(nz)sin(nx) ) dz (7) = 1 π ∫π −π f(z) ( 1 2 + ∑N ...

98 Chapter 1 Fourier Series and Integrals The addition formula for sines gives the equality sin (( N+^1 2 ) y ) =cos(Ny)sin ( 1 ...

1.7 Proof of Convergence 99 Under these conditions, the functionφ(y)of Eq. (16) has a jump disconti- nuity aty=0 and again is se ...

100 Chapter 1 Fourier Series and Integrals EXERCISES 1.Verify Lemma 2. Multiply through by 2 sin(^12 y). Use the identity sin ( ...

1.8 Numerical Determination of Fourier Coefficients 101 series is to be found for the function, some numerical technique must be ...

102 Chapter 1 Fourier Series and Integrals Figure 11 Preparation of a function for numerical integration of Fourier co- efficien ...

1.8 Numerical Determination of Fourier Coefficients 103 Ifris odd, Eqs. (3) and (4) are valid forn= 1 , 2 ,...,(r− 1 )/2, giving ...

104 Chapter 1 Fourier Series and Integrals ixi cosxi cos 2xi cos 3xi sin(xi)/xi 00 1. 01. 01. 01. 0 1 π 6 0. 86603 0 .50 0. 9549 ...

1.8 Numerical Determination of Fourier Coefficients 105 Figure 12 Graph of the difference betweenf(x)=sin(x)/xandF(x), the sum o ...

106 Chapter 1 Fourier Series and Integrals Jan. 2.751 July 3.861 Feb. 2.004 Aug. 4.088 Mar. 3.166 Sept. 4.093 Apr. 2.909 Oct. 3. ...

1.9 Fourier Integral 107 Now we modify Eq. (1). Letλn=nπ/aand define two functions Aa(λ)=^1 π ∫a −a f(x)cos(λx)dx, Ba(λ)=^1 π ∫a ...