Complex Analysis/Complex Numbers/Topology

As we have already seen the complex numbers are identified with the Euclidean Plane. So it is not surprising that much of what we know about the plane carries over to the complex numbers. In this section we will be specifically interested in topological properties of the complex plane. What are "topological properties"? In mathematics the term topology is used to describe certain geometric properties of spaces. Here we will mostly be concerned with ideas of open, closed, and connected. The notion of limits also falls under this section, because it is really a statement about the geometry of the complex plane to say two quantities are "close" or that one quantity "approaches" another.

We begin with the notion of a limit of a sequence of complex numbers.

We say that the limit of a sequence of complex numbers ${\displaystyle z_{n}}$ is ${\displaystyle z}$ if given ${\displaystyle \epsilon >0}$ there is a natural number ${\displaystyle N}$ so that if ${\displaystyle n>N}$ then ${\displaystyle |z-z_{n}|<\epsilon .}$

One difficulty with this notion of limit is that it requires us to know the limit ahead of time before we can decide if a sequence is convergent. To handle cases when we do not know the limit, an equivalent reformulation called the Cauchy criteria for convergence is stated below.

A sequence of complex numbers ${\displaystyle z_{n}}$ converges to some limit iff given ${\displaystyle \epsilon >0}$ there is a natural number ${\displaystyle N}$ so that if ${\displaystyle m,n>N}$ then ${\displaystyle |z_{m}-z_{n}|<\epsilon .}$

One direction of this equivalence is easy, but the other direction relies on the completeness of the Complex numbers. A topic we will defer until later.

Of course this is precisely the same as the definition for a limit of a sequence of points in ${\displaystyle \mathbb {R} ^{2}}$. Of course, a first important application of the limit of a sequence is defining convergence for a series of numbers.

Given a series ${\displaystyle \textstyle \sum _{n=1}^{\infty }z_{n}}$ we define partial sum of order ${\displaystyle N}$ to be the sum ${\displaystyle S_{N}=\sum _{n=1}^{N}z_{n}}$. We shall say the infinite series converges to ${\displaystyle z}$ if ${\displaystyle z=\lim _{n\to \infty }S_{n}}$.

Notice that when the Cauchy criteria is applied to infinite sums it takes the form: given ${\displaystyle \epsilon >0}$ there is an ${\displaystyle N}$ so that, if ${\displaystyle n,m>N}$ then ${\displaystyle {\bigg |}\sum _{m+1}^{n}z_{n}{\bigg |}<\epsilon }$.

To give a concrete example consider the series ${\displaystyle \sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}}$. For a fixed ${\displaystyle z}$ it can easily be seen that this series converges, and we shall denote the value it converges to by ${\displaystyle e^{z}}$.

To show this we simply we "bootstrap" from what we know about the real numbers. Recall that, for the real number ${\displaystyle |z|}$ we know that the sum for ${\displaystyle e^{|z|}}$ converges. Applying the Cauchy criteria to ${\displaystyle e^{|z|}}$ we know there is an ${\displaystyle N}$ so that, if ${\displaystyle n,m>N}$ then ${\displaystyle \sum _{m+1}^{n}{\frac {|z|^{n}}{n!}}<\epsilon }$. Now consider the series ${\displaystyle \textstyle \sum _{n=0}^{\infty }z^{n}/n!}$. For the ${\displaystyle N}$ determined above, lets examine ${\displaystyle {\big |}S_{n}-S_{m}{\big |}}$ for ${\displaystyle n\geq m>N}$.

${\displaystyle {\big |}S_{n}-S_{m}{\big |}={\bigg |}\sum _{m+1}^{n}{\frac {z^{n}}{n!}}{\bigg |}\leq \sum _{m+1}^{n}{\frac {|z|^{n}}{n!}}<\epsilon }$.

And hence by the Cauchy criteria the sum ${\displaystyle \sum _{n=0}^{\infty }z^{n}/n!}$ converges.

Even more, we showed the the series converges absolutely. Recall that for a series of real numbers, any possible reordering of the series converged to the same value. This theorem remains true for complex numbers. For us it will be very useful to examine the series ${\displaystyle e^{i\theta }}$ for theta a real number.

In this case we have

${\displaystyle e^{i\theta }=1+(i\theta )+{\frac {(i\theta )^{2}}{2}}+{\frac {(i\theta )^{3}}{6}}+{\frac {(i\theta )^{4}}{4!}}+{\frac {(i\theta )^{5}}{5!}}+{\frac {(i\theta )^{6}}{6!}}+{\frac {(i\theta )^{7}}{7!}}+\cdots }$

Now using that ${\displaystyle i^{2n}=(-1)^{n}}$ and ${\displaystyle i^{2n+1}=(-1)^{n}i}$ we can rewrite the series above as

${\displaystyle e^{i\theta }=1+i\theta -{\frac {\theta ^{2}}{2}}-i{\frac {\theta ^{3}}{6}}+{\frac {\theta ^{4}}{4!}}+i{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{6}}{6!}}-i{\frac {\theta ^{7}}{7!}}+\cdots }$

Finally if we rearrange the series to determine the real and imaginary parts we have that:

${\displaystyle e^{i\theta }={\bigg (}1-{\frac {\theta ^{2}}{2}}+{\frac {\theta ^{4}}{4!}}-{\frac {\theta ^{6}}{6!}}+\cdots {\bigg )}+i{\bigg (}\theta -{\frac {\theta ^{3}}{6}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots {\bigg )}}$

But now we notice by inspection that the series in the first set of parenthesis is exactly the Taylor series for ${\displaystyle \cos \theta }$ and the series in the second parenthesis is exactly the Taylor series for ${\displaystyle \sin \theta }$. And so we we conclude:

Euler's Formula
${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \!.}$

Thus we no longer need the name cis θ, we will instead simply use ${\displaystyle e^{i\theta }}$.

This section contains some more advanced topics that should perhaps be skipped on a first reading of this text.

Metric property

Define the metric ${\displaystyle d:\mathbb {C} ^{2}\to \mathbb {R} }$ as

${\displaystyle d(z_{1},z_{2})=|z_{1}-z_{2}|}$

It can easily be seen that ${\displaystyle d}$ satisfies positive definiteness, symmetry and the triangle inequality, implying that ${\displaystyle \mathbb {C} }$ is a metric space.

Completeness

Recall that a metric space is said to be complete if every Cauchy sequence converges to a limit.

For any point ${\displaystyle z_{0}\in \mathbb {C} }$, we call the open ball ${\displaystyle B_{\delta }(z_{0})}$, consisting of all the points ${\displaystyle z}$ such that ${\displaystyle |z-z_{0}|<\delta }$, a neighborhood of ${\displaystyle z_{0}}$. Similarly, a set consisting of points z such that ${\displaystyle |z|>\delta }$ for a positive δ will be called a neighborhood of infinity. Given a set ${\displaystyle {\mathfrak {G}}\subset \mathbb {C} }$, we call the set open if every point in ${\displaystyle {\mathfrak {G}}}$ has a neighborhood completely contained in ${\displaystyle {\mathfrak {G}}}$. Similarly, we call a set closed if its complement is open. A point ${\displaystyle z}$ is called an accumulation point of ${\displaystyle {\mathfrak {G}}}$ if every neighborhood of z contains a point in ${\displaystyle {\mathfrak {G}}}$ other than z itself. It can be shown that a set is closed if and only if it contains all of its accumulation points: see proof.

The Riemann Sphere

The Riemann Sphere

An interesting idea related to the extension of the complex numbers is the construction of the Riemann Sphere. The Riemann Sphere, essentially a stereographic projection, is constructed by projecting the Complex plane onto the unit sphere about the point ${\displaystyle (0,0,1)}$.

Formally, the rectangular coordinates of the projection ${\displaystyle (\xi ,\eta ,\zeta )}$ can be given by the transformations

${\displaystyle \xi ={\frac {z+{\bar {z}}}{1+z{\bar {z}}}},\eta ={\frac {1}{i}}{\frac {z-{\bar {z}}}{1+z{\bar {z}}}},\zeta =-{\frac {1-z{\bar {z}}}{1+z{\bar {z}}}}}$

Or equivalently, the reverse transformation,

${\displaystyle z={\frac {\xi +\eta i}{1-\zeta }}}$

The Riemann sphere is this transformation, together with the point ${\displaystyle (0,0,1)}$ labeled as ${\displaystyle \infty }$

It can also be shown that the stereographic projection preserves angles, and that circles and lines in the plane correspond to circles on the sphere: see proof.

In the metric |a-b| used earlier, the point z=∞ causes problems. However, using the stereographic projection, we can define another metric where the distance between two points a and b is the chordal distance

${\displaystyle \chi (a,b)={\frac {2|a-b|}{{\sqrt {1+a{\bar {a}}}}{\sqrt {1+b{\bar {b}}}}}}}$,

which has a well-defined meaning even when one of the points is ∞. We will only employ this metric when dealing with infinite values. For example, using this metric, neighborhoods of infinity do not require special treatment; we say that a neighborhood of a point ${\displaystyle z_{0}}$ is the set of all points z satisfying

${\displaystyle \chi (z,z_{0})<\delta }$,

where ${\displaystyle z_{0}}$ is allowed to be infinity.

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