# Complex Analysis

## Table of Contents

## Definitions

A **multivalued function** is a function which assigns to each point a *set of values* rather than a single value to each point, as a regular function would.

We define the **open ball** (or **open disk** since the complex plane is 2-dimensional) in the same way we do for the reals:

and the **closed disk**

Open and closed sets in the complex plane then follow the same definition as for *open* and *closed sets* in the 2-dimension real case.

### Functions

The *complex exponential function is the function defined by

and

Let be *non-zero*. We define the multivalued function by

We call *any* element of a **logarithm** of .

The **principal branch** of the logarithm function is the function defined by

for non-zero , where is the principal value of the argument function.

Or more general

The **branch cut** of a complex function is defined to be the point in the principal branch where the function is no longer holomorphic.

## Theorems

## Holomorphic functions

### Notation

- is almost always denoted
- , i.e. the
*unique*in the range - is the complex plane without origin => can be equipped with multiplication operation to form an (Abelian)
*group* *complex-valued function of a real variable*means , i.e. input is real and we can decompose the output into a real part and an imaginary part

### Differential Geometric view on `arg`

function

Let

- defined by , which is then surjective and continuous

Apparently, we call a *complex manifold* such as the one above a **Riemann surface**, and I honestly believe this is something I ought to return to once I have a better understanding of what complex manifolds are.

Observe then that this defines a fibre bundle, since for any two points we have the fibres be *isomorphic*, i.e. .

is the plane defined by Cartesian product since and . Thus, is in fact a topological manifold (probably equipped with the standard topology, but I haven't checked). Further we observe that is also a topological manifold. Thus, we our *fibre bundle* as claimed earlier.

### Complex-valued functions

Suppose we have a complex-valued function *of a real value*

We then say that is **continuous** at if and only if *both and are continuous at *.

If is instead a complex-valued function *of a complex variable*, let , then

We then say that is **continuous at ** if and only if for any there exists s.t.

where and are real valued functions, and this is thus equivalent of and being continuous at the point .

### Complex differentiability and holomorphicity

A function in a neighborhood of is **differentiable (everywhere in )** if and only if its real and imaginary parts and are continuously differentiable and obey the **Cauchy-Riemann equations** (everywhere in ):

*This can be seen by considering the "normal" definition of differentiability at a point, and observing the definition changes depending on which direction we approach the point from (vertical or horizontal direction).*

**Important:** it's crucial that the function be differentiable *in a neighborhood of * not just at the point !

We say that the complex-valued function is **holomorphic** at if it is differentiable everywhere in some neighbourhood of .

We say that a function is **entire** if it is *holomorphic in the whole complex plane*.

Let .

Then is **harmonic** if it satisfies the *Laplace equation*:

Let be open, and let be harmonic.

We say a harmonic function is the **harmonic conjugate of ** if the complex-valued function

is holomorpic on .

#### Solving differential equations

This techinque can be used to solve differential equations on "ugly" domains.

Consider the following:

### Polynomials and rational functions

### Complex powers

Let with . Then we define ** th power of ** by

Unless stated otherwise, a complex power is defined in the principal branch.

Let with . Then

- if there is exactly one value of
- if where are
*coprime*, with there are exactly values of - if or , there are
*infinitely*manu values of

Let . Then the values

where

are the **roots of unity**.

### Graphing complex functions

- Hard to visualize as 2D + 2D becomes 4D
- Good idea to treat each of the different 3D plots separately

Let be *open*, and let .

We say is **conformal** if preserves angles: i.e. if the angle between the images under of two straight lines in are equal to the angle between the two straight lines themselves.

By applying this definition to tangents of differentiable curves, more generally we can say the same about the angles between curves at certain points.

We're saying the push-forward of , is *angle-preserving*.

Therefore, any diffeomorphism between and is *angle-preserving*.

Consider . For , . Then we consider the surface

and the function

defined by .

- This is the
*projection*of a bundle; further, forms a fibre bundle (right?)

Let be open, and let be holomorphic.

Then preserves angles at every where .

This theorem is useful for establishing the image of a function (if it's a polytope (?)) since we can simply compute the mapped values at the edges (i.e. curves from vertex to vertex) and, knowing that the angles are preserved, immediately know how the edges between the mapped vertices look.

## Möbius transformations

A **Möbius transformation** is a function of the form

where are such that .

Observe that is *not* defined on the entirety of , which leads us to defining the extended complex plane.

*We'll often consider the case , if nothing else is specified.*

That is, a Mobius transformation is defined on .

The **extended complex plane** is the set , where is just some object .

We extend the usual arithmetic operations in the following way: for and non-zero ,

Consider the coordinates describing . We identify the complex plane with the plane defined by , and a complex number with the point .

The **Riemann sphere** is the unit sphere in defined by

and we consider the "north pole" to be the point .

The Riemann sphere therefore has two charts:

- For all points in the complex plane, the
*chart*is the identity map from the sphere (with removed) to the complex plane. - For , the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending to and all other points to .

Let such that the three points , , and are *colinear*.

It is clear that

thus we define also and thereby consider as being defined on the extended complex plane, i.e. .

The map is evidently *bijective*, so it has an inverse . This function is the **sterographic projection**.

*Better description:*

The unit sphere in is the set of pints such that .

Let be the "north pole", and let be the rest of the sphere.

The plane (xy-plane) runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.

For any point on , there is a *unique line through and *, and this line intersects the plane in *exactly* one point . We define the **stereographic projection** of to be this point in the plane.

and

Stereographic projection maps a circle to either a circle or a straight line (a "circline").

Makes some people say that *straight lines* are *circles of infinite radius*.

A

**translation**is a Möbius transformation of the formwhich corresponds to the matrix

A

**rotation**is a Möbius transformation of the formso that for some , which corresponds to the matrix

A

**dilation**is a Möbius transformation of the formwhich corresponds to the matrix

An

**inversion**is a Möbius transformation of the formwhich corresponds to the matrix

We say that a Möbius transformation ** fixes the point of infinity** if .

Translations, rotatons, and dilations fix the point at infinity, while inversions do not.

Let be a Möbius transformation. Then is a composition of a finite number of translations, rotations, diluations, and, if and only if does *not* fix the point at infinity, *one* inversion.

Möbius transformations map circlines to circlines.

Let be three distinct points. Then there exists a unqiue Möbius transformation such that

This is useful because we by simply knowing how the Möbius transformation maps the three different points, we can tell what it does to circles or lines.

Let be *distinct* points.

The **cross-ratio** of the four points is the image of under the Möbius transformation which sends to .

## Complex integration

### Complex integrals

Let be a interval, and of the form

Then is **integrable** if its real and imaginary parts are integrable in the usual (real) sense, and we define the integral of by

*It will usually suffice to observe that continuous functions are integrable.*

#### Example

where denotes the arc-length of .

- is by def. closed and bounded, i.e. a compact set
- is
*continuous*

Hence is in fact *bounded* i.e. finite.

### Contour integrals

Let be distinct.

Then a *(parametrized) curve * connecting and is a continuous function

Writing and , we decompose into a real and imaginary parts for continuous real functions , so

We say the curve is *regular* if is continuously differentiable and forall .

A curve from to in is a **contour** if it is a finite union of regualr curves, which together joint with , i.e. there exists

such that

such that is *continuous*.

For a continuous function , we define the **contour integral of along ** by

Let be a curve in , and let be continuous. Then

Let . We will say that is a **domain** if is open and every two points in can be connected by a contour which lies wholly on .

Let be a domain, and be *continous*. Then the following are equivalent:

- has an antiderviative on
- for all closed contours in
- all contour integrals are independent of path , and depend only on the endpoints.

### Cauchy's Integral Theorem

Let be a contour in .

Then is *simple* if it has no self-intersections, except possible at the endpoints, i.e. for all distinct , unless and and is a closed contour.

A **loop** is a simple, closed contour.

Let be a loop in . Then defines two regions in the , with as their common boundary:

- a boundary domain, the
*interior*of - an unbounded domain, the
*exterior*of

Let be a loop in . We say is **positively-oriented** if as we move along the curve in the direction of parametrization, the interior is on the LHS.

*Otherwise state, all loops should be assumed to be positively-oriented.*

is said to be **simply connected** if the interior of every loop on lies wholly in .

Let and be a loop in which does not pass through . Then

Since , we clearly have two cases:

- : integral is zero due to Cauchy Integral theorem
- : Consider the following figure:
Since this figure basically "encapsulates" all possible loops which have
*outside*!

We break the integrand , aswell as the differential into their real and imaginary components:

In this case we have

By Green's Theorem, we may then replace the integrals around the closed contour with an area integral throughout the domain that is enclosed by as follows:

and for imaginary part,

However, being the real and imaginary parts of a function holomorphin in the domain , and must satisfy the Cauchy-Riemann equations there:

We therefore find that both integrads (and hence their integrals) are zero:

Which gives us the result

as wanted.

Let be a loop, , and be holomorphic inside and on . Then

Let be a loop, be holomorphic inside and on , and lie inside .

Then is infinitively differentiable at and, for all positive integers ,

### Liouville's Theorem and its applications

Let be a domain, and be such that , be holomorphic on , and be such that for all .

Then for all , we have that

### Maximum Modulus Principle

and be such that the closed disc , and be holomorphic on .

Then

Let:

- be holomorphic on a domain
- s.t.

Fix , then

where is the circle of radius centred at (Cauchy Integral formula).

is parametrized by , given by

So, using the definition of the contour integral

Suppose

i.e. maximum value is attained at the center of the circle. Then,

hence

which, since is clearly constant, we can use the linearity of the integral to get

which is a non-negative continuous function, which implies that the expression *inside* the integral is zero for all , i.e.

which is true for every point in is of this form, for some , and some .

Hence is constant on .

Let , and be holomorphic and bounded on , i.e.

for some .

If achives its maximum at , then is constant on .

## Series Expansions for Holomorphic Functions

### Stuff

Let be a power series.

Then s.t.

- converges if
- converges uniformly on for all
- diverges if

is holomorphic on its disc of convergence , where is the radius of convergence.

To proof holomorphicity of convergent power series, one could do as follows:

Fix . Then for some . Series converges uniformly on , thus partial sums are holomorphic, which implies series is holomorphic at .

Let be open, and .

Then we say that is **analytic** if at everypoint , can be expressed as a convergent power series.

Suppose is holomorphic on . Then the Taylor series

converges to on , and converges uniformly on for all

where

for any loop with .

It's REALLY important to realize that this Taylor series is only on some *disk* !!!

Therefore, if we know is holomorphic on some open subset , then if we want to talk about the Taylor series on any point , still can only say something about this **on some disk **!!

The Taylor series at some point is NOT necessarily the same as the Taylor series at some *other* point .

Let be the circle of radius centered at where . By CIF we know that for we have

So

where we've used the fact that

thus

hence we could recognize the fraction above as the convergent series. Further, due to the series being convergent, we can interchange the summation and integration

which is just the **Taylor series**!

Taylor series of a (if it exists) is *unique*, i.e. if is holomorphic on and

A anulus is

Suppose is holomorphic on the anulus .

Then

for any loop with for all , and the series converges uniformly on for all .

We say is a **singularity** if is not holomorphic at .

Suppose has a singularity at .

If there exists s.t. is holomorphic on the punctured disk , has a Laurent series centred at , valid on this disk

Then ONE of the following is true:

- for all ( is a
**removable singularity**) - such that but for all ( is a
**pole of order**) - infinitively many negative s.t. ( is an
**essential singularity**)

If you have a removable singularity and then consider the integral along some loop around .

Then the integral is the integral along some loop on which is holomorphic, hence it's zero.

We can therefore redefine to take on the value at the singularity point , and we got ourselves a holomorphic function on the non-punctured disk centered at !

is a **zero of holomorphic function ** if .

Further, we say it's a **zero of order ** if

but

Then

Let , be a neighbourhood of , be holomorphic on , and such that for a sequence of distinct points which converge to .

Then is identically zero on some disc centered at .

Let be holomorphic at be a zero of of order .

Then has

- has a pole of order at if
- has a pole of order (if ) at , if is a zero of order of , OR is a removable singularity if

### Analytic continuation

Suppose is holomorphic on a domain with and .

Then

Power series above *might* have radius of convergence

is a holomorphic on with , i.e.

One might then as *does on *?

**Yes**, is an analytic continuation of to where

and is well-defined.

Let be domain, and be holomorphic.

We say that a holomorphic function is an **analytic continuation** of if

Let be a domain, , and be holomorpic on and such that

for a sequence of distinct points which converge to . Then

Let be a domain, , and be holomorphic on and such that

for a sequence of distinct points which converge to . Then

Let be holomorphic at , where is zero of order . Then

- if is not zero of , then has a pole of order at
- if is a zero of order of , then has a pole of order at if , and has a removable singularity at otherwise.

## Cauchy Residue

### Theorem

Let , be holomorphic on the punctured disc for some , with an isolated singularity at , and be a loop *inside* , with . Then

where is the coefficient of the term in the Laurent expansion of centred at ,

Let and be holomorphic on the *punctured* disc , for some , with an isolated singularity at .

Then the **residue** of at , is

where is the term in the Laurent series of centered at .

Let , and be holomorphic on the punctured dist for some , with removable singularity at . Then

Let and be holomorphic on the *punctured* disc , for some , with a pole of order at . Then

Let and , be holomorphic on for some , such that has a *simple zero* () at , while . Then, defining we have

Let be a loop, and be holomorphic *inside* and *on* except for finitely many isolated singularities . Then

Let be a domain.

A function is **meromorphic** on if for all , either has a pole at or is holomorphic at .

### Application: trigonometric integrals

Integrals of the form

for *ration function* , can often be evaluated by considering a contour integral of appropriate function around the unit circle centered at .

On with we have

and

Defining , we therefore have that

If parametrized by defined by , then

It basically comes down to rewriting and with , which then *often* provides us with a *rational function* of which it's substantially esaier to obtain the *singularities*, thus the *residue*, hence the integral around .

### Improper Integrals

We define the **Cauchy principal value** of the integral

as

Let be a rational function, where

ad such that . Then

where and are the semicircular contours from to in the upper and lower half-plane, respectively.

Let

- be a domain
- , be meromorphic on with a simple pole at
- be the circular arc parametrized by for for some .

Then

Let , and let be a loop with . Then