References:

  • topology: wikipedia
  • complex analysis: Evan Chen, The Napkin

topology

  • compact: every open cover has a finite sub-cover.

  • complete: limits of cauchy sequences exist

  • open set (in a metric space): \(\forall x\in U, \exists \varepsilon>0\) such that \(\left\{ y\; : \;d(x,y)<\varepsilon \right\}\subseteq U\).

  • discrete topology: all sets are open

    • is compact iff finite

  • product topology: minimal amount of open sets to make projections \(\pi_i : \prod_j X_j \mapsto X_i\) continuous for each \(i\).

    • continuous: \(f^{-1}(\text{open set}) =\text{open set}\)

  • open sets in the product topology are unions of sets of the form \[\prod_i U_i\]

    • where \(U_i\) is open in \(X_i\) and \(U_i\neq X_i\) for only finitely many \(i\).

  • Tychonoff’s theorem: product of compact spaces is compact

complex analysis

Derivative: \[\lim_{h\to 0} \frac{f(x+h)-f(x)}{h},\] if it exists, which is a high bar because \(h\to 0\) means complex limit.

holomorphic / entire: complex differentiable on domain / all of \(\mathbb{C}\).

ex: polynomials, \(\exp\), nice compositions of holomorphic functions.

Countour integrals

\[\oint_\alpha f(z) dz = \int_a^{b} f(\alpha(t)) \alpha'(t) dt\]

Theorem. Let \(\gamma\) parameterize the unit circle. Then \[\oint_\gamma \frac{1}{z} dz = 2\pi i.\]

For any integer \(m\neq -1\), \[\oint_\gamma z^{m} dz = 0.\]

Theorem. \(\gamma\) a loop, \(f\) holomorphic

\[\oint_\gamma f(z) dz = 0.\]

Another way of saying this is, if you have two different curves with the same endpoints then the contour integrals will be the same.

Theorem. Let \(D\) be a disk, and \(\gamma\) parameterize the boundary \(t\mapsto R\exp(it)\). Let \(f\) be holomorphic. For any \(a\) in the interior of \(D\) we have

\[f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a} dz.\]

Theorem. Over a disk, a holomorphic function is given exactly by its Taylor Series.

Some important results:

Theorem. Let \(f\) be entire, and suppose \(|f(z)|< 1000\) for all \(z\). Then \(f\) is the constant function.

memomorphic functions

  • holomorphic: proto-ex: polynomials
  • memomorphic: proto-ex: rational functions

memomorphic functions are basically holomorphic everywhere except for a couple of “poles”.

Definition.

memomorphic function: not identically zero on any open neighborhood.

so basically just has a bunch of isolated poles.

Another way of defining memomorphic functions is in terms of Laurent Series:

For a function to be memormophic it should be holomorphic in neighborhoods of all points except for a “small” number of poles. (this is allowed to be infinite, e.g., like \(1/\sin(z)\) is memomorphic) but the poles must have non-intersecting open balls around them). If \(a\) is a pole then it is required to, in some neighborhood of \(a\) admit a series expansion \[f(z) = \frac{c_{-m}}{(z-a)^{m}} + \cdots + c_0 + c_1 (z-a) + \cdots .\]

So it’s like a Taylor series but we can have some pole crap at the front.

In the expression \(\frac{c}{z^{m}}\) \(c\) is callled the residue, \(m\) the order of the pole (simple pole is \(m=1\))

Definition.

Winding number: Let \(\gamma\) be a loop.

\[\frac{1}{2\pi i}\oint_\gamma \frac{1}{z-p} dz\]

Theorem.

\[\frac{1}{2\pi i}\oint_\gamma f(z) dz = \sum_{\text{pole } p} \text{winding}(\gamma, p) Res(f; p)\]