How important is the assumption $\gamma$ is positively oriented? (Residues, Cauchy's Thm from Cauchy's Integral Formula)

Question

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.32, Cor 8.27

Question 1. Should the following 2 statements in the textbook have an assumption that the path $\gamma$ in question is positively oriented?

Question 2. Are there ways to forego assuming $\gamma$ is positively oriented? Eg 'If $\gamma$ is simple, piecewise smooth and closed but not positively oriented, then $-\gamma$ is' or something. This may be a Calculus III issue.

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2 statements:

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1. (Cor 8.27) Suppose $f$ is a function that is holomorphic in $A = \{R_1 < |z-z_0| < R_2\}$ with Laurent series $$f(z) = \sum_{k=-\infty}^{\infty} c_k (z-z_0)^k$$ If $\gamma$ is any simple, closed, piecewise smooth, path in $A$ s.t. $z_0 \in int(\gamma)$, $$\int_{\gamma} f = 2\pi i c_{-1}$$

• Reason: At the start of Ch9.2 on Residues, the text restates Cor 8.27 but with an assumption that $\gamma$ is positively oriented.

2. (Exer 4.32) Show that the corollary (Cor 4.20) to Cauchy's Thm (Thm 4.18) is a corollary to Cauchy's Integral Formula (Formula 4.27) if $\gamma$ is simple.

• Reason: Cauchy's Integral Formula (Formula 4.27) assumes $\gamma$ is positively oriented while Cor 4.20 doesn't.

Answer 1

If $-\gamma$ is the reverse of contour $\gamma$, $\oint_{-\gamma} f = - \oint_\gamma f$. So the formula of Cor. 8.27 can't be true for both $\gamma$ and $-\gamma$: you do need to assume $\gamma$ is positively oriented.

On the other hand, Cor. 4.20 says $\oint_\gamma f = 0$: if that's true for $\gamma$, it's also true for $-\gamma$, so this does not need to assume positively oriented.