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Math Help - Complex Variable Hard Question

  1. #1
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    Thumbs down Complex Variable Hard Question

    Could somebody please help me with the following question?


    Let Γ be the path with parametrization
    f(z)=1/2iz-1
    (i) Describe the effect that f has on a typical point of C in terms of geometric transformations

    Let Γ(t) be the path with parametrization

    γ(t)=1+2i+2*sqrt2*e^it (t is between [1/4pi,3/4pi])

    (ii)Sketch Γ indicating the direction of increasing t,and identifying is initial and final points in cartesian form.

    (iii) Sketch the path f( Γ ) by applying the geometric transfrormation from the first part to Γ, showing the effect of wach transformation in turn. Indicate the direction of f( Γ )and identify its initial and final points in cartesian form.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint :

    Consider the transformation f:\mathbb{C}\to \mathbb {C},\;f(z)=az+b

    and h,r,t:\mathbb{C}\to \mathbb {C} defined by:

     h(z)=|a|z\;\textrm{(homothecy)\;},\quad r(z)=e^{i\arg a}z\;\textrm{(rotation)\;},\quad t(z)=z+b\;\textrm{(translation)\;}

    Then,

    f=t\circ r\circ h .
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