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\title{Mathematics 311:  Complex Analysis --- Assignment 1}
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{\bf Reading}:  Marsden, sections 1.1, 1.2, 1.4.

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{\bf Problems}:

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1.  Show that $\C$, the field of complex numbers, is isomorphic to the set
$$
S=\left\{\rmat{x}{-y}yx:x,y\in\R\right\}
$$
with the usual matrix addition and multiplication.  (An isomorphism is an
oper\-a\-tion-preserving bijection.)

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2.  (a) Find the two values of $\sqrt{-9i}$;
(b) Find the four values of $\root4\of{-16}$.  Express your answers
(though not your intermediate work) in Cartesian form.

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3.  A valid position of minute and hour hands on a clock is called 
{\em exchangeable\/} if swapping the hands yields another valid clock position.
Find all exchangeable clock positions.  
(The idea is to use complex numbers, of course.)

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4.  Show that if $a$, $b$ are complex numbers with $b\ne0$ then 
$a+\overline{a}+b\overline{z}+\overline{b}z=0$ is the equation of a
line in the plane, and any line in the plane is described by such an
equation.  (The equation of a line is $A+Bx+Cy=0$ where not both $B$
and~$C$ are zero.)

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5.  Show that all roots of the polynomial $z^8+2iz^5+4$ lie outside the unit
circle.  (The idea is to use the Triangle Inequality.)

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6.  By evaluating the product $(2+i)(3+i)$ and then viewing the result
geometrically, show that
$$
\pi/4 = \arctan1/2 + \arctan1/3.
$$

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7.  Any three complex numbers can be taken as the vertices of triangle, 
so one can refer to such a triangle as the triangle $(z_1, z_2, z_3)$.

(a) Show that the triangles $(z_1, z_2, z_3)$ and $(w_1, w_2, w_3)$ are 
similar with $z_j$ corresponding to $w_j$ for $j=1,2,3$ if and only if
$$
\det\left[\begin{matrix}
1 & 1 & 1 \\ z_1 & z_2 & z_3 \\ w_1 & w_2 & w_3
\end{matrix}\right] = 0.
$$
(Hint: both conditions hold if and only if some affine map $z\mapsto az+b$ 
takes $z_j\mapsto w_j$ for $j=1,2,3$.)

(b) Use part (a) to show that the triangle $(z_1, z_2, z_3)$ is equilateral
if and only if
$$
z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1.
$$

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