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Therefore u ( x , y) = x 2 + y 2 So while u = 2x x We have v v = =0 x y So unfortunately, the Cauchy-Riemann equations cannot be satis ed, unless x = y = 0 We conclude the function is not analytic Another way to look at this is that the function has z dependence: f ( zz ) = z 0 = z z (unless of course, z = 0) This illustrates the fact that a function which depends on z is not analytic and u = 2y y v ( x , y) = 0

onto the top-most empty menu placeholder in the menu and movie list When you do, Sonic DVDit! will add an additional menu placeholder at the bottom of the list

World Wide Web Kerberos Remote Telnet Service Post Office Protocol Version 2 Post Office Protocol Version 3 Network News Transfer Protocol Network Time Protocol NetBIOS Name Server NetBIOS Datagram Service NetBIOS Session Service News Simple Network Management Protocol Simple Network Management Protocol Traps Border Gateway Protocol Secure HTTP Remote Login Talk

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SOLUTION We write the function in terms of real and imaginary parts as e z = e x +iy = e x eiy Now use Euler s formula to write eiy = cos y + i sin y So we have e z = e x (cos y + i sin y) = e x cos y + ie x sin y Therefore u( x , y) = e x cos y v ( x , y) = e x sin y

5 Be sure to frequently select the File/Save command from the Sonic DVDit!

We nd the following partial derivatives of these functions: u = e x cos y x v = e x sin y x u = e x sin y y v = e x cos y s y

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menu as you are working on your movie to save any changes you have made When creating a menu, a Photos To Go logo will appear outside the TV safe zone, so it will not show up when your movie is played on a TV If you wish to purchase menu backgrounds from Photos To Go without the logo, visit their Web site by using the Photos To Go link in DVDit! s Help menu At this point you have two options for creating the buttons for your menu: As I mentioned earlier, you can drag-and-drop movie clips from the media palette onto the menu background in the viewer window to create a menu button which links to the first frame of the selected clip The button will use the first frame of the clip as a thumbnail You can open the button palette and then drag-and-drop a button onto the menu background This new button will initially not be linked to anything, so you ll have to add your links manually Fortunately, adding the links manually is quite easy

Since ux = v y and uy = v x, the Cauchy-Riemann equations are satis ed, and we conclude the function is analytic

Figure 13-3 shows an example of a menu that has four buttons Three of them were created by dragging-and-dropping video clips onto the menu The fourth (the lower-left button) was created by dragging-and-dropping a menu button onto the menu This fourth button will need to have a link added manually Notice that you are not locked into a specific menu layout in the way you are in the template-based editors

If the partial derivatives ux , u y , v x , and v y are continuous at a point ( x 0 , y0 ) and the Cauchy-Riemann equations hold then f ( z0 ) = ux ( x 0 , y0 ) + iv x ( x 0 , y0 ) f ( z0 ) = v y ( x 0 , y0 ) iu y ( x 0 , y0 ) (324)

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It follows from the Cauchy-Riemann equations that we can write f df f = i y dz x (325)

You can make a number of changes to the menus that you create in Sonic DVDit! Because your menus are not based on a pre-built template, you have great flexibility in creating just the menu appearance you want In fact, the only real limitation on your menus is that they cannot contain more than 36 buttons on a single menu I doubt that you will consider this to be much of a limitation! Because Sonic DVDit! gives you so much flexibility, it s hard to imagine everything you might do with your menus Let s take a look at some of the things that you are likely to want to do with them

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