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\noindent{\Large{\textsc{Complex Variables \hfill Assignment 9.1  \\ Fall 2024 \hfill Due November 6}}}

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\begin{exercise}
Let $\varnothing \ne A \subset \C$ be open, let $f:A \to \C$ be continuous, and let $\gamma:[a,b] \to \C$ be a piecewise smooth path whose image lies in $A$.  If $a = t_0 < t_1 < t_2 < \cdots < t_n = b$ and $\gamma$ is smooth on each $[t_{j-1},t_j]$, we defined
\[
\int_{\gamma} f(z) \, dz =  \int_a^b f(\gamma(t)) \, \gamma'(t)  \, dt = \sum_{j=1}^n \int_{t_{j-1}}^{t_j} f(\gamma(t)) \, \gamma'(t).
\]
Prove that this is independent of the partition $P = \{ t_j \}$ of $\gamma$ into smooth pieces.  [{\em Suggestion}:  If $P$ and $P'$ are both partitions of $[a,b]$ on whose subintervals $\gamma$ is smooth, show that the integral using either partition is equal to the integral using the common refinement $P'' = P \cup P'$.]
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\begin{exercise}
Textbook exercise 2.1.1.
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\begin{exercise}
Textbook exercise 2.1.7.
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