Thread: 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.

Hint :

Consider the transformation

and defined by:



Then,

.