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  \textbf{\color{darkblue}MAT 103A: Complex Analysis (W21)}\quad
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  \fbox{\parbox{5.5in}{
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    \textbf{\large  Your Name: } 
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 \noindent Please review the problem set guidelines in the \href{https://people.ucsc.edu/~jbreland/teaching/W21_MATH103A/build/W21_103A_syllabus.pdf}{syllabus} before turning in your work. Note: you may not use any results proved in class on 2/2/21 or 2/4/21 to solve these problems (i.e. anything after Section 46 of the textbook cannot be used). 
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\item \begin{enumerate}[(a)]
	\item Let $w = \cos^{-1} z$. Show that $\cos^{-1} z = -i \log(z+i(1-z^2)^{1/2})$ by solving the equation 
	\[
	z = \frac{e^{iw} + e^{-iw}}{2}.
	\]
	
	\item The expression for $\cos^{-1} z$ is multiple-valued. By taking the principal branch of $\log$ and the principle branch of the square root\footnote{Evidently, the value of $\frac{d}{dz} \cos^{-1} z$ depends on the branch of square root we choose.}  (making $\cos^{-1} z$ single-valued and analytic), show that
	\[
	\frac{d}{dz} \cos^{-1} z = \frac{-1}{(1-z^2)^{1/2}}.
	\]
	
\end{enumerate}



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\item Let $z:[a,b] \to \C$ be a parameterization of a smooth arc $C$ and suppose $f(z)$ is analytic at a point $z_0 = z(t_0)$ on $C$. Show that\footnote{Hint: Write $f(z) = u(x,y) + iv(x,y)$ and $z(t) = x(t) + iy(t)$. Then $(f\circ z)(t) = u(x(t),y(t)) + i v(x(t),y(t))$. Use the chain rule from vector calculus to compute $(f\circ z)'$ and then use the Cauchy-Riemann equations.}
\[
(f\circ z)'(t) = f'(z(t))z'(t) 
\]
when $t=t_0$. 


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\item Let $\alpha, \beta \in \R$. Evaluate the following integral of real-valued functions
\[
\begin{array}{ccc}
	\ds \int_0^\pi e^{\alpha x}\cos \beta x \, dx & \text{and} & \ds \int_0^\pi e^{\alpha x}\sin \beta x \, dx
\end{array}
\]
simultaneously by computing a \emph{single} integral of a complex-valued function. \footnote{Hint: consider the complex-valued function of a real variable $f(x) = e^{\alpha x}\cos \beta x + ie^{\alpha x}\sin \beta x$.}


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\item Let $z_1, z_2\in \C$. Compute the integral 
\[
\int_C 1 \, dz
\]
where $C$ is any contour joining $z_1$ to $z_2$. 


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\item Let $C$ denote the unit circle with positive orientation. Compute the integral
\[
\frac{1}{2\pi i} \int_C \frac{(1+z)^n}{z^{k+1}}\, dz
\]
for any integers $0 \leq k \leq n$.\footnote{Hint: you may assume the binomial theorem holds for complex numbers.}

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\item Integrate the function $f(z) = \conj{z}$ over the following contours:
\begin{enumerate}[(a)]
	\item $C_1$: the line segment joining $0$ to $1+i$;
	
	\item $C_2$: the line segment joining $0$ to $1$, following by the line segment joining $1$ to $1+i$. 
	
\end{enumerate}
	
	
\end{enumerate}


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