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\noindent{\Large{\textsc{Complex Variables \hfill Assignment 7.2  \\ Fall 2024 \hfill Due October 16}}}

\large

\vspace{1cm}

\begin{exercise}
Let $\gamma : [a,b] \to \C$ be a path.  We say $\gamma$ is {\em differentiable} at $t_0 \in (a,b)$ if
\[
\gamma'(t_0) = \lim_{t \to t_0} \frac{\gamma(t) - \gamma(t_0)}{t-t_0}
\]  
exists.  If we write $\gamma(t) = x(t) + i \, y(t)$, it is easy to show that $\gamma$ is differentiable at $t_0$ if and only if the (real-variable/real-valued) functions $x(t)$ and $y(t)$ are, and in this case
\[
\gamma'(t) = x'(t) + i y'(t).
\]
Use the multivariate chain rule (for real-valued functions) and the Cauchy-Riemann equations to prove that if $A \subseteq \C$ is open, $f:A \to \C$ is analytic on $A$, and $\gamma : [a,b] \to A$ is a differentiable path in $A$, then $f \circ \gamma: [a,b] \to \C$ is a path in the range of $f$, and the chain rule holds:
\[
\frac{d}{dt} f(\gamma(t)) = f'(\gamma(t)) \cdot \gamma'(t) \,\,\, \text{ for all }  t \in (a,b).
\]
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\begin{exercise}
Textbook exercise 1.5.17.
\end{exercise}

\begin{exercise}
Textbook exercise 1.5.20.
\end{exercise}



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