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FUNCTIONS OF A COMPLEX VARIABLE x sin x 2 e dx 2 1 2 4 x sin x 2e 3 dx 4e x x2 1 2 cos x dx . cosh x 2 cosh =2 Hint: Consider ! Then let R ! 1: 1 16.84.

The RegistryRootKey parameter stores the browser window s size and location in the Windows Registry. This lets your application make the Web browser appear with the same size and location each time it is called. The information is stored in HKEY_CURRENT_USER and so is specific to each user. If you don t want to use this capability, pass a null (empty) string in this parameter. The OpenButtonAlwaysEnabled parameter enables URL link selection. If True, the Select button is enabled at all times, and the browser can return any URL, whether it s an actual file or just a link. Use this if your program just needs to obtain a URL from the user and doesn t need to actually open or work with the contents of a remote file. The following example uses LaunchBrowserDialog to get a URL from the user and displays the selection in a message box. Figure 7-4 illustrates this code in action. Public Sub TestLaunchBrowserDialog() Dim strStartUrl As String Dim strInput As String Dim blnStatus As Boolean strStartUrl = InputBox("Enter a URL", , "http://www.apress.com")

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= F(b) F(a) =

1 16.83.

x 2 + 1 x by making the substitution u = x 2 + 1. Then, du = 2x dx, and 1 1 u du = 2 2 u1/2 du = 1 u3/2 2 3

1 16.85.

1 1 1 = u3/2 = (x 2 + 1)3/2 = ( x 2 + 1)3 3 3 3 Hence, by the fundamental theorem,

1 16.86.

The following code snippet uses a message box to inform you of the active space for the current document: If ThisDrawing.ActiveSpace = acModelSpace Then MsgBox "The active space is model space" Else MsgBox "The active space is paper space" End If To change the active space, you need to set the value of the ActiveSpace property. The following code contains a macro you can use to toggle between the two spaces: Public Sub ToggleSpace() With ThisDrawing If .ActiveSpace = acModelSpace Then .ActiveSpace = acPaperSpace Else .ActiveSpace = acModelSpace End If End With End Sub

1 1 1 ( x 2 + 1)3 = (( 12 + 1)3 ( 02 + 1)3 ) 3 3 0 3 1 3 1 = (( 2) ( 1) ) = (2 2 1) 3 3

MISCELLANEOUS PROBLEMS 16.87. If z ei and f z u ; i v ; , where and are polar coordinates, show that the Cauchy-Riemann equations are @u 1 @v ; @ @ @v 1 @u @ @

( 2)3 = ( 2)2 2 = 2 2

16.88. If w f z , where f z is analytic, de nes a transformation from the z plane to the w plane where z x iy and w u iv, prove that the Jacobian of the transformation is given by @ u; v j f 0 z j2 @ x; y 16.89. Let F x; y be transformed to G u; v by the transformation w f z . Show that if all points where f 0 z 6 0, @2 G @2 G 0. @u2 @v2 @2 F @2 F 0, then at @x2 @y2

( 1)3 = 13 = 1

az b , where ad bc 6 0, circles in the z plane are trans16.90. Show that by the bilinear transformation w cz d formed into circles of the w plane. 16.91. If f z is analytic inside and on the circle jz aj R, prove Cauchy s inequality, namely j f n a j @ where j f z j @ M on the circle. n!M Rn

Alternatively, you can simply use the following line of code: ThisDrawing.ActiveSpace = (ThisDrawing.ActiveSpace + 1) Mod 2

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