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432 Advanced Calculus, Fifth EditionEXAMPLE Let f (x) = ex. Then, for a = 0 and x > 0,Henceexn+'O
I .Chapter 6 Infinite Series 433EXAMPLE 2t:Since coshx = ;(ex + P), one findsEXAMPLE 3 Since sin x cos x = 5 sin 2x, one findssinxcosx = - - - - +...+21! 23x3 3!(- 1)"-122"-l*Zn-1(2n - l)!I+ . .} . (6.53)Remark. Since the remainder formula does provide a method for estimating R,, it 'enables one to estimate the error committed in computing the sum of a power series.Thus if f (x) is known to be analytic in a given interval and it is known that in thisinterval,for a certain constant Mn+1, then : .This formula can be added to those developed in Section 6.9.As Theorem 41 shows, one does not require analyticity of the function f (x) inorder to be able to apply the remainder formula; f (x) need only have continuousderivatives through the (n + 1)st order. Actually, the (n + 1)st derivative need onlyexist between a and x, continuity not being required. Thus in principle, one can use ' 'the formula as a method for evaluating a nonanalytic f (x) by a Jinite series, withremainder estimated by (6.54).Four other operations by which new Taylor series can be obtained are described bythe following theorems.THEOREM 42Convergent power series can be multiplied; that is, ifmM-Ti
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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 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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256 Advanced Calculus. Fifth Editio
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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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Chapter 5 Vector Integral' Calculus
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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 7Fourier Series and Orthogo
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y the Schwarz inequality (7.46). By
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IiChapter 7 Fourier Series and Orth
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Functions of aComplex VariableWe as
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Chapter 8 Functions of a Complex Va
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TChapter 8 Functions of a Complex V
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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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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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Partial DifferentialEquationsA part
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Chapter 10Partial Differential Equa
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Equations (10.66) and (10.67) then
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For t = 0 the series (10.99), if co
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These are easily solved for A and B
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Accordingly,Chapter 10 Partial Diff
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Section 10.12, page 6941 - (-1)"1.
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