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If n 2 and f z 5z2 3z 2, then f 00 1 10. Hence, 2! 5z2 3z 2 5z2 3z 2 dz or dz 10 i 10 3 2 i C z 1 z 1 3 C Method 2. 5z2 3z 2 5 z 1 2 7 z 1 4. Then 5z2 3z 2 5 z 1 2 7 z 1 4 dz dz 3 z 1 z 1 3 C C d dz dz 7 4 5 2 i 7 0 4 0 5 2 3 z 1 C C z 1 C z 1 10 i

to indicate that f (x) does get larger without bound. EXAMPLE Let f (x) = 1/x2 for all x = 0. The graph of f is shown in Fig. 9-2. As x approaches 0 from either side, 1/x2 grows

(b) Since jzj 10 encloses both poles z 1 and z 3 the required integral 2 i e 5e 3 16 16 ! i e 5e 3 8

without bound. Hence,

EVALUATION OF DEFINITE INTEGRALS 16.27. If j f z j @ M for z Rei , where k > 1 and M are constants, prove that lim R!1 Rk where is the semicircular arc of radius R shown in Fig. 16-9.

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'' get the point, length, height, and angle from user With ThisDrawing.Utility '' get point, length, height, and angle from user varpnt1 = .GetPoint(, vbCr & "Pick the start point: ") dblLength = .GetDistance(varpnt1, vbCr & "Enter the length: ") dblHeight = .GetDistance(varpnt1, vbCr & "Enter the height: ") dblAngle = .GetAngle(varpnt1, vbCr & "Enter the angle: ") '' calculate remaining rectangle points dbl90Deg = .AngleToReal("90d", acDegrees) varpnt2 = .PolarPoint(varpnt1, dblAngle, dblLength) varpnt3 = .PolarPoint(varpnt2, dblAngle + dbl90Deg, dblHeight) varpnt4 = .PolarPoint(varpnt3, dblAngle + (dbl90Deg * 2), dblLength) End With '' draw the rectangle With ThisDrawing .ModelSpace.AddLine .ModelSpace.AddLine .ModelSpace.AddLine .ModelSpace.AddLine End With End Sub 8 covers creating drawing objects.

By the result (4), Page 394, we have M M f z dz @ j f z jjdzj @ k R k 1 R R

The notation lim f (x) = shall mean that f (x) gets smaller without bound as x approaches a; that is,

R!1 R!1

1 2 x

Fig. 16-9

=

varpnt1, varpnt2, varpnt3, varpnt4,

z Re ,

Sometimes the value f (x) will get larger without bound or smaller without bound as x approaches a from one side (x a or x a+ ).

1 1 1 2 @ @ 4 if R is large enough (say R > 2, for 4 e4i 1 R jR4 e4i j 1 R4 1 R example) so that M 2; k 4. Note that we have made use of the inequality jz1 z2 j A jz1 j jz2 j with z1 R4 e4i and z2 1.

CHAP. 9]

dz , where C is the closed contour of Problem 16.27 consisting of the line from R to R z4 1 and the semicircle , traversed in the positive (counterclockwise) sense. Since z4 1 0 when z e i=4 ; e3 i=4 ; e5 i=4 ; e7 i=4 , these are simple poles of 1= z4 1 . Only the poles e i=4 and e3 i=4 lie within C. Then using L Hospital s rule, & ' 1 Residue at e i=4 lim z e i=4 4 z 1 z!e i=4 1 1 3 i=4 e lim 3 4 z!e i=4 4z & ' 1 3 i=4 Residue at e lim z e3 i=4 4 z 1 z!e3 i=4 1 1 9 i=4 e lim 3 4 z!e3 i=4 4z Consider

Use the TranslateCoordinates method to translate a point from one coordinate system to another. This is often necessary when the user is working in a construction plane, for example. Most AutoCAD input and object-creation methods work exclusively in WCS, so you need to translate user input to correctly work in the construction plane. This method returns an array of doubles. It has the following syntax: dblAngle = UtilityObject.AngleFromXAxis(Point, From, To, Displacement [, Norm]) Table 7-17 explains this method s parameters.

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CHAP. 16]

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