Thread: Complex Variable Hard Question

1. Complex Variable Hard 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.

2. Hint :

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

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

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

Then,

$\displaystyle f=t\circ r\circ h$ .

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