Question

The rational function p(2) = 1 (2-1)4 + 4 is holomorphic on the domain C\{Q1, 22, 23, 24} for some four distinct points 21, 22, 23, and 04. (a) Find the values of a1, 22, 23, and 04. [8] (b) Use one of the Cauchy's formulas to evaluate the integral of p(2) along y, a positively oriented closed contour parametrising a rect- angle with vertices Fi and 4 £i. Show that this integral can be expressed in the form of (a +ib) where a, b and c are integers. [8]

Answer 1

Answer (a): The given function is analytic everywhere except at the poles given by setting denominator equal to zero as follows:

$(z-i)^4+4=0\\ \implies z-i=4^{1/4}(-1)^{1/4}\\ \implies z-i=4^{1/4}e^{(2n+1)i\pi/4}; n=0,1,2,3\\ {\color{blue}(\because -1=\cos(2n+1)\pi+i\sin(2n+1)\pi=e^{i(2n+1)\pi})}\\ \implies z=i+4^{1/4}e^{(2n+1)i\pi/4}; n=0,1,2,3\\ \color{red} a_1=i+4^{1/4}e^{i\pi/4}\\ a_2=i+4^{1/4}e^{i3\pi/4}\\ a_3=i+4^{1/4}e^{i5\pi/4}\\ a_4=i+4^{1/4}e^{i7\pi/4}\\$

(b) Observe that only simple pole $a_4=i+4^{1/4}e^{i7\pi/4}$ will lie inside the given rectangle hence

$\oint p(z)dz=2\pi i\times Res[p(z),z=a_4]\\ \implies \oint p(z)dz=2\pi i\times \left (\frac{1}{4(z-i)^3}\right )_{z=a_4}=\color{magenta}\frac{\pi i}{2}4^{-1/4}e^{-7\pi i/4}$