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 are equivalent to the following PDE: $df/dx + i df/dy = 0$ You might want to use this fact in the problems below, though it's not necessary. - 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; conclude that any polynomial function is holomorphic. - Show that the product of two holomorphic functions 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?