Show that I dz Jc (z2 – 1)2 + 3 + 2 where C is the positively oriented boundary of the rectangle whose sides lie along the lines x = +2, y = 0, and y = 1
Suggestion: By observing that the four zeros of the polynomial q(z) = (z2 – 1)2 + 3 are the square roots of the numbers 1 + V3i, show that the reciprocal 1/q(z) is analytic inside and on C except at the points 13+i - and - Zo=- -V3+i zo = 12 Then apply Theorem 2 in Sec. 76. Theorem 2. Let two functions p and q be analytic at a point zo. If p(zo) = 0, 9(20) = 0, and q'(zo) #0, then zo is a simple pole of the quotient p(z)/q(z) and Res P(z) _ p(zo) og (z) g' (zo): (2)