1. Write the formula for multiplication of complex numbers in rectangular coordinates.  How does this relate to the ``angle sum'' formulae from trigonometry?

- Use De Moivre's theorem to write down a ``triple angle'' formulae, i.e. closed form expressions for $\sin 3x$ and $\cos 3x$.

- Show that every nonzero complex number has exactly $3$ cube roots.  What are the cube roots of $i$?  Draw them in the complex plane.

- Show that the Cauchy-Riemann equations for $f=u+iv$ are equivalent to the following PDE:

      $\frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} = 0$

  You might want to use this fact in the problems below, though it's not necessary.

5. Write down the Cauchy-Riemann equations in polar coordinates.

- 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?

-  Show that the sum of two holomorphic functions is holomorphic.

-  Show that the product of two holomorphic functions is holomorphic

-  Conclude that any polynomial function is holomorphic.

-  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? 
    a.  $\sinh(z), \cosh(z)$
    -  $\frac{z^3}{1 + z^2}$
    -  $\sin(z), \cos(z)$
    -  $\sqrt{z}$
    -  $\log z$
    -  $\mathrm{erf}(z)$, 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?

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