Please could you give me a general method for sketching the loci made by things such as
arg (z+1) = pi/12 etc.
And I have the transformation
w = (z - 1) / z
and can transform that into z = w-1 / (w – 1)
but how do I show that it maps |z-1| = 1 in the z-plane onto |w| = |w – 1| in the w-plane?
And how do find the region T to shade on an Argand diagram when
The region |z – 1| < 1 in the z-plane is mapped onto the region T in the w-plane.
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Zs and Ws and Argand diagrams... P6 Q watch
- Thread Starter
- 27-06-2005 08:45
- 27-06-2005 08:57
well i can do the first part
arg(z) = pi/12 is a half line going at angle pi/12 from the origin, and then use regular transformations for arg(z+1) as it will just have its start point 1 to the left
i fink with the second part you take one from both sides
z = w-1 / (w – 1)
z-1 = (w-1 / (w – 1)) - 1
z-1 = (w-1 / (w – 1)) - (w – 1)/(w – 1)
Then rearrange to make one fraction on the RHS
Then mod both sides and replace the |z-1| with 1 and then multiply out to get rid of the fractions
For the last bit if you put in the < signs during your working for the last part and sub in z = x + iy you should be able to get a cartesian equation
- 27-06-2005 09:10
The first part is just a translation. f(z-a) is simply the locus for z but translated by the complex number 'a'.
You might want to check your expression for z. You can then set |z-1|=1 and form a common denominator.
For the next part you could choose a point in the region in the z plane and see which side of the line in the w plane it maps to.
(Original post by Gaz031 in his sig)
- 27-06-2005 09:16
Why are people ashamed of being illiterate but proud of being innumerate?