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-## Countability
+1. Write the formula for multiplication of complex numbers in rectangular coordinates.  How does this relate to the ``angle sum'' formulae from trigonometry?
 
-1. Group the following sets according to their cardinality:
+- Use De Moivre's theorem to write down a ``triple angle'' formulae, i.e. closed form expressions for $\sin 3x$ and $\cos 3x$.
 
-    a. $\mathbb{N} = \{ 1,2,3,4,\dots \}$
-    - $\mathbb{Z} = \{ \dots, -2, -1,0,1,2, \dots \}$
-    - $\mathbb{N} \times \mathbb{N}$
-    - $\mathbb{Q}$ = Set of all fractions $\frac{n}{m}$ where $n,m \in \mathbb{Z}$
-    - $\mathbb{R}$
-    - The open interval $(0,1)$
-    - The closed interval $[0,1]$
-    - $2^{\mathbb{N}}$ = Set of all subsets of $\mathbb{N}$.
-    - $2^{\mathbb{R}}$ = Set of all subsets of $\mathbb{R}$.
-    - $\mathbb{R}^{\mathbb{R}}$ = Set of all functions from $\mathbb{R}$ to itself.
+- Show that every nonzero complex number has exactly $3$ cube roots.  What are the cube roots of $i$?  Draw them in the complex plane.
 
-Cook up other examples and post them on the wiki!
+- Show that the Cauchy-Riemann equations are equivalent to the following PDE:
 
-2. Let $X$ be any set.  Show that the cardinality of $2^{X}$ is larger than the cardinality of $X$.  
-(Hint: Let $f: X \to 2^X$ be a bijection.  Consider the set of all elements $x \in X$ such that $x$ is not an element of $f(x)$.)
+$df/dx + i df/dy = 0$
 
+You might want to use this fact in the problems below, though it's not necessary.
 
-## Fourier Series
+- Show that the function $f(z) = \overline{z}$ is not holomorphic, despite being angle-preserving.  How does this function transform the complex plane?
 
+- Show that the function $f(z) = z^n$ is holomorphic for any integer n (possibly negative!).  How do these functions transform the complex plane?
 
-1. Compute the Fourier Series of the following functions.  Do both the exponential and sin/cos expansions.
-    a. $f(x) = \sin^3(3x)\cos^2(4x)$
-    - $g(x) = x(x-2\pi)$  
-      (Hint: Use integration by parts)
+-  Show that the sum of two holomorphic functions is holomorphic; conclude that any polynomial function is holomorphic.
 
-2. Show that  
-$\int_0^{2\pi} \sin^4(x) dx = \frac{3 \pi}{4}$  
-(Hint: write out the exponential fourier expansion of $\sin^4(x)$.)
+-  Show that the product of two holomorphic functions is holomorphic.
 
-3. Compute the exponential Fourier coefficients of $\sin^2(x)$:  
-$a_n = \frac{1}{\sqrt 2\pi} \int_0^{2\pi} \sin^2(x) e^{-inx} dx$  
-and use this to show that  
-$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$  
+-  Try to extend the following functions of a real variable to holomorphic functions defined on the entire complex plane.  Is it always possible to do so?  What goes wrong? 
+  -  $\sinh(z), \cosh(z)$
+  -  $\frac{z^3}{1 + z^2}$
+  -  $\sin(z), \cos(z)$
+  -  $\sqrt{z}$
+  -  $\log z$
+  -  $\mathrm{erf}(z)$, the antiderivative of the gaussian $e^{-z^2/2}$
+  -  $e^{1/z}$
+What is the growth rate of the magnitude of these functions as $z \to \infty$ along the real axis?  The imaginary axis?  How does the argument change?