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252 Advanced Calculus, Fifth EditionA second type of improper integral, generalizing the definite integrals withinfinite limits, is an integralJ/f(X,Y)dXdYRwhere R is an unbounded closed region. Here one obtains a value by a limit processjust like that of (4.9 1). The most important case of this is that of a function continuousoutside and on a circle x2 + y2 = a2. If f (x, y) is of one sign, the integral over thisregion R can be defined as the limit-?where Rk is the region a2 5 x2 + y2 ( k2. Thus the improper integral 4has the value12rk2-P - a 2-~lim lk - r: dB r dr = lim 2nk+ w k-+m 2 - ~ 'which equals 2 ~ a ~ -- ~ 2), / for ( ~ p > 2. For p 5 2 the integral diverges.While the emphasis here has been on double integrals, the statements hold withminor changes [affecting in particular the critical value of p for the integral (4.92)]for triple and other multiple integrals.For further discussion of this topic, see Section 6.26.4PROBLEMS1. One way of evaluating the error integralis to use the equations& e-"*dx 3Z(lme-.' dx)l = dm ePx' dx law e-y2dy = JoW Lm7 ,e-x--v-dx dyand to evaluate the double integral by polar coordinates. Carry out this evaluation, showingthat the integral equals fi; also discuss the significance of the above equations in termsof the limit definitions of the improper integrals.'2. Show that the integral// log ~ m ddyxRconverges, where R is the region x2 + Y2 5 1, and find its value. This can be interpretedas minus the logarithmic potential, at the origin, of a uniform mass distribution over thecircle.J4
3. a) Show that the integralChapter 4 Integral Calculus of Functions of Several Variablesover the spherical region x2 + y2 + z2 5 1 converges for p
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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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Differential Calculusof Functionsof
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Chapter 2 Differential Calculus of
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PROBLEMSChapter 2 Differential Calc
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defined analogously:Chapter 2 Diffe
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magnitudeand the force is inversely
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Page 279 and 280: Vector Integral CalculusPART I.TWO-
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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 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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Chapter 6 Infinite Series 461b) Let
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Therefore R3 lies in some annulus h
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Fourier Series andOrthogonal Functi
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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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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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Chapter 10Partial Differential Equa
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Equations (10.66) and (10.67) then
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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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