Today I Learned

Some of the things I've learned every day since Oct 10, 2016

171: More General Cauchy Integral Theorem

Let $\Omega \subseteq \mathbb{C}$ be open, and let $\Gamma \subset \Omega$ be a cycle which is null homologous. (That is, $\Gamma$ is a collection of closed curves the winding number of which is 0.) Then

$\int _\Gamma f(z) dz = 0$.

154: Morera’s Theorem

In complex analysis, Morera’s Theorem states that if $D \subset \mathbb{C}$ is a region (open, path-connected) and $f: D \rightarrow \mathbb{C}$ is continuous and has the property that

$\int _\gamma f(z) dz = 0$

for any closed curve $\gamma \subset D$, then $f$ is holomorphic.

This is an immediate corollary to the observation in [153], since if $F$ is a primitive of $f$ on $D$ then $f$ itself is holomorphic on $D$.

153: Sufficient Condition for Existence of Primitive of Complex Function

If $\Omega \subset \mathbb{C}$ is open and path-connected, and a continuous function

$f: \Omega \rightarrow \mathbb{C}$

has the property that

$\int _\gamma f(z) dz = 0$

for any closed curve $\gamma \subset \Omega$, then $f$ has a primitive (antiderivative) on $\Omega$.

(The converse is obviously true, since if $F$ is a primitive of $f$ then for any closed curve $\gamma$ which begins and ends at a point $z$, $\int _\gamma f(z) dz = F(z) - F(z) = 0$.)

147: Cauchy Integral Formula

Suppose $\Omega \subset \mathbb{C}$ is open, $f: \Omega \rightarrow \mathbb{C}$ be holomorphic, and $C \subset \Omega$ be the boundary of a disk as a curve with positive (counterclockwise) orientation. Then where $z$ belongs to the interior of this disk,

$f(z) = \frac{1}{2 \pi i} \int _C \frac{f(w)}{w - z} dw$.

That is, the value of $f$ on the interior of the disk is completely determined by its values on the boundary of the disk. This is really cool!

145: Cauchy’s Theorem (Complex Analysis)

Let $\Omega$ be an open subset of $\mathbb{C}$, $\gamma \subset \Omega$ be a closed curve whose interior is contained in $\Omega$, and $f: \Omega \rightarrow \mathbb{C}$ be holomorphic. Then Cauchy’s Theorem states that

$\int _{\gamma} f(z) dz = 0.$

143: Complex Integrals Over Curves With Identical Sets but Opposite Orientation

Given a curve $\gamma$, let $\gamma ^{-}$ be the curve which covers the same subset of $\mathbb{C}$ as $\gamma$, but has opposite orientation (goes in the other direction). Then

$\int_{\gamma^{-}} f(z) dz = - \int_{\gamma} f(z) dz$,

assuming the integral over $\gamma$ exists in the first place. So reversing the direction of a curve just reverses the sign of an integral over that curve.

138: Projected Circles on the Reimann Sphere

Under the Reimann Sphere projection, circles on the sphere are mapped to “circles” in the extended complex plane $\mathbb{C}^*$.

The quotes are because every line in $\mathbb{C}^*$ is mapped to by a circle on the Reimann Sphere passing through the north pole $\mathcal{N}$. So in this view, lines in $\mathbb{C}$ are just circles in $\mathbb{C}^*$.

137: Continuity of Complex Functions at Isolated Points

Complex functions are continuous at isolated points. That’s pretty much it.

More formally, let $z_0 \in \Omega \subset \mathbb{C}$ be isolated, so that for some real $r$

$D_r (z_0) \cap \Omega = \{z_0\}$.

Then where $f: \Omega \rightarrow \mathbb{C}$, $f$ is continuous at $z_0$.

Intuitively this just because being within a certain distance of $z_0$ implies that a a point is $z_0$ itself, which means the point will map under $f$ to the same place as $z_0$.

136: The Reimann Sphere

The Reimann Sphere is a way of setting up a bijection between points on a 2-sphere and the extended complex plane ($\mathbb{C}$ plus the “point at infinity”).

From ‘Complex Analysis’ by Stein and Shakarchi

Take a 2-sphere $\mathbb{S}$ with diameter 1, set it on the origin of $\mathbb{C}$, and let $\mathcal{N}$ be the ‘north pole’ of $\mathbb{S}$. Given a point $W$ on $\mathbb{S}$, let $w$ be the point in where $\mathbb{C}$ is hit by the line between$\mathcal{N}$ and $W$. This is called the stereographic projection of $W$.

This mapping gives you a bijection between points $W$ on $\mathbb{S}$ and points $w$ in the extended complex plane, with $\mathcal{N}$ corresponding to the point at infinity.

In viewing the extended complex plane as simply a 2-sphere, the point at infinity is on equal footing with all other points in $\mathbb{C}$, whereas before it had to be given special treatment.

103: Dedekind Cuts

The Dedekind cut is a method of constructing $\mathbb{R}$ from $\mathbb{Q}$. A cut is first defined to be a partition of $\mathbb{Q}$ into 2 non-empty subsets $A, B$ such that

1. if $p \in A$ and $q < p$, then $q \in A$
2. if $p \in A$, then there is a $q \in A$ such that $q > p$ (meaning $A$ has no maximum element)

A cut can be equivalently determined solely by $A$ alone, rather than the pair $(A, B)$.

Cuts can then be used to construct $\mathbb{R}$ by defining any $r \in \mathbb{R}$ to be the cut where $A$ is the set of all members $q$ of $\mathbb{Q}$ such that $q < r$. That is, $r$ is simply defined as the subset of rationals smaller than itself, and $\mathbb{R}$ is the set of all such subsets.