Show that f is conformal describe the geometric effect of f

Question

(A) Find the derivative of the function

f(z) = Log(2-iz)/z³,

and determine the domain of this derivative.

(B) (i) Use the definition of the derivative of a complex function to prove that the function

f(z) = 3z^- + 2i

is not differentiable at 1 + i.

(ii) Use the Cauchy-Riemann equations to find all the points at which the function

f(x+iy) = x² + y³ + 2ixy

is differentiable, and find its derivative at these points.

(C) The function f is defined by

f(z) = e^πiz-1.

(i) Show that f is conformal.

(ii) Describe the geometric effect of f on a small disc centred at 3/2.

(iii) Show that the paths

Γ₁: γ₁(t) = 2√3 + 2e^it    (t ∈ [0, 2π]),

Γ₂: γ₂(t) = (√3 - i)t       (t ∈ R),

are smooth and meet at the point 2√3 - 2i.

(iv) Determine the angle from the path f(Γ₁) to the path f(Γ₂) at the point f(2√3 - 2i).