Reference no: EM131316346
(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) = eπiz-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 + 2eit (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).
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