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    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.


    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

    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)
    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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