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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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