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488 Advanced Calculus, Fifth Editionfor an appropriate positive integer N. Since the functions sinn x and cosn x areexpressible as trigonometric polynomials (Problem 8 following Section 7.4), thefunction P(x) is a trigonometric polynomial. LetThen k > 1 and P 2 kN for (x- xol 5 i6. Since $ is positive (greater than 1) forIS 2 5 Ix - xo I < 6, P is positive in this range. On the other hand, I $(x)l < 1 for-n 4x
Chapter 7 Fourier Series and Orthogonal Functions 489Proof. Let the"3um of the Fourier series of f (x) be denoted by fi (x):Since the series converges uniformly, it follows from Theorem 1 that fi(x) is continuousand that a,, bn are the Fourier coefficients of fi (x). But the series was given asthe Fourier series of f (x). Hence f (x) and f, (x) have the same Fourier coefficients,and by Theorem 3, f (x) = fl(x); that is, f (x) is the sum of its Fourier series for-n Fx_
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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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Advanced Calculus. Fifth Editionhas
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Advanced Calculus, Fifth Edition8.
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Advanced Calculus, Fifth EditionBy
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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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*I. 421 8 Advanced Calculus, Fifth
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Advanced Calculus, Fifth EditionFig
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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 379If a s
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Chapter 6 Infinite Series 381Every
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Chapter 6 Infinite Series 383PROBLE
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Chapter 6 Infinite Series 385EXAMPL
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Chapter 6 Infinite Seriesi ,Figure
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n Chapter 6 Infinite Series 391If L
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Chapter 6 Infinite Series 393of con
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EXAMPLE 8zz, 5. Again the ratio tes
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12. Determine convergence or diverg
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The theorem has a counterpart in te
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Chapter 6 Infinite Series 401Proof.
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at. Chapter 6 Infinite Series 407Fi
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F; Chapter 6 .Infinite Series 41 1C
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for when x = $, the numerator Ixn 1
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EXAMPLE 1Ezl 5. Here the ratio test
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. ,. Chapter 6 Infinite Series 423D
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, Chapter 6 Infinite Series 425by T
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Chapter 6 Infinite Series 427THEORE
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nL: Chapter 6 Infinite Series 429ha
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PROBLEMSChapter 8 Functions of a Co
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2f 'c) J, zeZ dz on the line segmen
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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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Ordinary DifferentialEquations9.1 D
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For t = 0 the series (10.99), if co
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