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# Zs and Ws and Argand diagrams... P6 Q watch

1. 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.

??
2. 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
3. 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.
4. (Original post by Gaz031 in his sig)
Why are people ashamed of being illiterate but proud of being innumerate?
i'd be kinda t'other road about... if i was more innumerate than i am i'd be mucho gutted, but my lack of literary prowess does not bother me. who needs words (other than pi, induction, set etc) when the language of the universe is numbers?

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