Further Analysis | Part IB, 2004

Let CC be the contour that goes once round the boundary of the square

{z:1Rez1,1Imz1}\{z:-1 \leqslant \operatorname{Re} z \leqslant 1,-1 \leqslant \operatorname{Im} z \leqslant 1\}

in an anticlockwise direction. What is Cdzz\int_{C} \frac{d z}{z} ? Briefly justify your answer.

Explain why the integrals along each of the four edges of the square are equal.

Deduce that 11dt1+t2=π2\int_{-1}^{1} \frac{d t}{1+t^{2}}=\frac{\pi}{2}.

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