a) Let f(z) = z^2 and γ(t) = 1 + it^3, t ∈ [0,1].

Question

i) Write out the contour integral ∫γ f(z)dz as an integral with respect to t. You do not need to evaluate this integral.

ii) Evaluate the integral ∫0,1+i z^2dz

iii) What is the relationship between the integrals in (i) and (ii)? Give reasons for your answer.

b) Use Green's Theorem to evaluate the integral
∮C (5 + 10xy + y^2)dx + (6xy + 5y^2)dy
where C is the rectangle with vertices (0, 0), (0, a), (b,a) and (b, 0) traversed counterclockwise.

c) Consider the contour γR(t) = Re^it, t ∈ [0, pi]. Show that
∫γR dz/(z^2 + 4) -> 0, as R -> infinity.

Answer 1

so, where do you get stuck? Surely your text discusses this topic.

Answer 2

just wanted solutions and maybe working out as my previous year exams papers don't come with solutions. So I cant compare to see if I got it correct or not.