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\title{Mathematics 311:  Complex Analysis --- Assignment 2}
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{\bf Reading}:  Marsden, sections 1.3, 1.5, 1.6.

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{\bf Problems}:

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1.  Show that if $f(z)$ is analytic in a region $\Omega$ and either
$\text{Re}f(z)$ or $|f(z)|$ is constant in $\Omega$ then $f(z)$ must be 
constant there.

\medskip

2.  Show that if $f(z)$ is analytic and its second partial derivatives exist 
and are continuous then
$\Delta(|f(z)|^2)=4|f'(z)|^2$, where $\Delta$ is the Laplacian operator, 
$\Delta=\partial^2/\partial x^2+\partial^2/\partial y^2$.

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3.  Show that the function
$$
f(z)=\begin{cases}
e^{-1/z^4}&\qquad\text{for }z\ne0\\
0&\qquad\text{for }z=0
\end{cases}
$$
is analytic at all $z\ne0$, is not analytic at $z=0$, but satisfies the 
Cauchy--Riemann equations at $z=0$.  
(For $z\ne0$, decompose $f$ as a composition of functions each known to
be analytic; to show $f$ is not analytic at $0$ it suffices to show $f$ is not
even continuous at $0$; the form $f_x=-if_y$ of the Cauchy--Riemann equations 
is easiest to check at $0$.)

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4.  Extend the trigonometric functions to complex arguments by defining
\begin{align*}
\sin z &= \sin x \cosh y + i \cos x \sinh y \\
\cos z &= \cos x \cosh y - i \sin x \sinh y
\end{align*}
for all $z=x+iy\in\C$; here the hyperbolic functions are defined as usual by
$\sinh y = (e^y - e^{-y})/2$, $\cosh y = (e^y + e^{-y})/2$ for $y\in\R$.

(a) Show that $\sin z$ is analytic for all $z\in\C$ and find all points $z$ 
for which $\sin z=0$.

(b) Show:  $\cos(\pi/2-z)=\sin z$, $\cos(z+\pi)=-\cos z$, 
$\cos(z+2\pi)=\cos z$, $\sin^2z+\cos^2z=1$, $\cos z=(e^{iz}+e^{-iz})/2$ 
for all $z\in\C$.

(c) Discuss the mapping described by the function $w=\cos z$.
It suffices to consider the strip $0\le\text{Re}z\le2\pi$, by periodicity.  
The cosine function can be written as the composition of
the functions $z\to iz$, $z\to e^z$, $z\to(z+z^{-1})/2$, each of which is
familiar.  Illustrate suitable restrictions of each of these functions to give
a good sense of the composite.  On what part of the strip is $\cos$ 1-to-1? 
What is its output? 

(d) Discuss a single-valued inverse cosine function:  where it is defined, 
a formula for it in terms of the logarithm function, 
what its branch points are.

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