Today I Learned

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

Category Archives: analysis

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.

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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”).

reimann

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.