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The Cauchy-Riemann equations can also be derived by looking at limits We do this by writing everything in terms of real and imaginary parts to that once again we take f ( z ) = u( x , y) + iv ( x , y), z 0 = x 0 + iy0 , and z = x + i y Now following what we did earlier and taking w = f ( z ) for notational convenience, we have w = f ( z0 + z ) f ( z0 ) = u( x 0 + x , y0 + y) u( x 0 , y0 ) + i[ v ( x 0 + x , y0 + y) v ( x 0 , y0 )] If the function f ( z ) is differentiable, we can approach the origin in any way we like So we try this in two ways, rst along the x axis (and hence setting y = 0) and then along the y axis (and setting x = 0) Going with the rst case, we set y = 0 and get w u( x 0 + x , y0 ) u( x 0 , y0 ) [ v ( x 0 + x , y0 ) v ( x 0 , y0 )] = +i z x x Now, taking the limit x 0 we see that these expressions are nothing other than partial derivatives That is:

You use the Cutting Room tool to set the frames at which a clip begins or ends playing You can also use this tool to extract a single frame as a bitmap image, to strip out the audio track, or (in some cases) to strip out the video track The Cutting Room tool also enables you to select the frame that will be used as the thumbnail and to split a clip into two or more pieces To use the Cutting Room tool, click the Cutting Room button on the Roxio VideoWave Power Edition toolbar When you do, the playback controls below the viewer window will add controls so you can mark the in-point and the out-point, as shown here (these are the controls that sit below the scrubber) The in-point is the frame where the clip begins playing, and the out-point is the frame where it stops playing

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To extract a part of the clip, set the thumbnail, or split the clip at the current frame, you use the controls at the lower-right side of the Roxio VideoWave Power Edition window, as shown here Click the Apply button when you have made your changes, and the Close button when you are finished using the Cutting Room tool

[ v ( x 0 + x , y0 ) v ( x 0 , y0 )] u( x 0 + x , y0 ) u( x 0 , y0 ) w + i lim = lim x 0 x 0 x z x v u +i = x x

If the derivative f ( z ) exists, we must obtain the same answer even if we choose another way for ( x , y) to go to zero Now we use the other option, approaching the origin along the y axis Hence we set x = 0 This time we have w u( x 0 , y0 + y) u( x 0 , y0 ) [ v ( x 0 , y0 + y) v ( x 0 , y0 )] = + z i y y

If you want to make several splits in a scene, you will probably find that using the Scene Detector mentioned earlier in this chapter is somewhat easier to use than the Cutting Room tool

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