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494 Advanced Calculus, Fifth EditionThen fl (x) and f2(x) can be represented by Fourier series:n 2 O0 cos(2n - l)x sin nxf2(x) = --- C1)" -, -n
Chapter 7 Fourier Series and Orthogonal Functions 495b) Show that if a > 0 and a 5 1x1 5 n, then Ipn(x)ls 1/ sin la.C) Show that the nth partial sum Sn(x) of the seriessin 2x sin nxsinx + - +.-.+-+... \,2 n, can be written as followsPI(X) +Sn(x)= - -+...+p2(x)pn-l(x)+-.pn(x)1.2 2.3 n(n - 1) n[Hint: Write sin x = pl , sin 2x = p2 - pl, and so on.]d) Show that the series Cgl --,sinnxis . uniformly convergent for a 5 (x I 5 n, where a > 0.[Hint: By (c),Hence uniform convergence of the sequences S,* and pn/n implies uniform convergenceof Sn(x). The sequence S,* converges uniformly, since it is the nth partial sumof the series2 P~(x) 'n=l n(n + 1)'which converges uniformly, because of (b), by the M-test. The sequence pn/n convergesuniformly to 0 by (b).]If one reviews the theory of expansion of a function in a Fourier series, as presentedin the preceding sections of this chapter, it is natural to ask why the trigonometricfunctions sin nx and cos nx play such a special part and whether these functionscan be replaced by other functions. If one is interested only in periodic functions,there is indeed no natural alternative. However, if one is concerned with representationof a function over a given interval, it will be seen that a great variety ofother series is available, in particular, series of Legendre polynomials, Bessel functions,Laguerre polynomials, Jacobi polynomials, Hermite polynomials, and generalSturm-Liouville series.Let f (x) be given in an interval a ( x ( b; this interval will be fixed throughoutthe following discussion. Let ~$~(x), ~$~(x), ..., C$n(x), ... be functions all piecewisecontinuous in this interval; this sequence is to replace the system of sines and cosines.We then postulate a development,just as in the case of Fourier series. Our next step, for Fourier series, was to multiplyboth sides by cos mx or sin mx and integrate from -n to n; when we did this, allterms dropped out except the one in a, or b,, respectively, because of the relations
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ContentsVectors and Matrices1.1 Int
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3.6 Combined Operations 183*3.7 Cur
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*6.24 Principal Value of Improper I
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Case of Two Particles 662Case of N
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Differential Calculusof Functionsof
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Chapter 2 Differential Calculus of
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PROBLEMSChapter 2 Differential Calc
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Chapter 2 Differential ~alculus of
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defined analogously:Chapter 2 Diffe
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magnitudeand the force is inversely
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Vector Integral CalculusPART I.TWO-
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Chapter 5 Vector Integral Calculus
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3. Evaluate the following line inte
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5. Evaluate by Green's theorem:a) $
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Green's theorem now givesChapter 5
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*5.15 PHYSICAL APPLICATIONSChapter
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Infinite Series6.1 INTRODUCTIONAn i
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Chapter 6 Infinite Series 377These
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Chapter 6 Infinite Series 383PROBLE
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Chapter 6 Infinite Series 385EXAMPL
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n Chapter 6 Infinite Series 391If L
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EXAMPLE 8zz, 5. Again the ratio tes
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12. Determine convergence or diverg
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Chapter 6 Infinite Series 401Proof.
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for when x = $, the numerator Ixn 1
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nL: Chapter 6 Infinite Series 429ha
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PROBLEMSChapter 8 Functions of a Co
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Rule IV could also be used, with A
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RULE V If f (z) has a zero of first
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+aJ 12. Evaluate the following inte
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PassibleChapter 8 Functions of a Co
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C) the first quadrant of the w plan
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Ordinary DifferentialEquations9.1 D
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These are easily solved for A and B
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