What closed paths make $\int_{\gamma} \frac 1 z dz = 2m \pi i$

cauchy-integral-formulacomplex integrationcomplex-analysiscontour-integrationproof-verification

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.14

This text talks about winding numbers in Def 1.17. In the textbook, I think Exer 4.13 refers to winding numbers.

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  1. Does Exer 4.13 refer to winding numbers, and is the $m$ in Exer 4.14 a winding number of $\gamma$?

  2. Can I choose unit circle but keep it winding $m-1$ more times, i.e. $\gamma(t) = e^{it}, t \in [0,2 m\pi]$? I mean it doesn't say simple $\gamma$.

  3. Actually, how about $\gamma=\gamma_1\gamma_2 \cdots \gamma_m$ where $\gamma_2, \dots, \gamma_m$ are copies of a closed simple path $\gamma_1$ s.t. $0 \in int(\gamma_1)$?

Best Answer

In 4.13, for $n = -1$ you have the winding number of $\gamma$ about the origin. An immediate reason the others are zero is that for $n \not= -1$, $z^n$ is a derivative.

For your second question, the answer is yes.

For your third question, the answer is __ BCLC: yes I guess? ___.