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Complex Analysis Notes

1. Complex Numbers

Matrix Representation

Proof of Triangle Inequality

Thus:

Properties of Complex Conjugate

Argand Diagrams

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Circle

Line

Set Concepts

Open Set: A set is open if every point of the set is an interior point (it contains no boundary

points), meaning that at every point in the set there is some open disk that is completely

contained in the set.

Closed Set: A set is closed if it contains all of its boundary points, where a boundary point is any

point about which every open disk contains at least one point in the set and one point outside

the set.

Closed and Open: A set is closed if and only if its complement is open. Clopen sets are both open

and closed. The only two in C are all of C and the empty set. Sets that are neither open nor

closed are simple: just include some but not all of the boundary points.

Bounded Set: A set is bounded if every element of the set has a smaller absolute value than

some number R.

Compact Set: A compact set is one which is bounded and closed.

Simply Connected Set: Basically this means the set does not have 'holes'.

A sphere is simply connected because every loop can be contracted (on the surface) to a point

A torus is not simply connected. Neither of the colored loops can be contracted to a point

without leaving the surface.

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2. Series and Convergence

Convergence in C

A complex sequence converges to if for any there is an such that:

Cauchy Convergence

A cauchy sequence is one where for any there is an such that:

A complex sequence converges iff it is cauchy. Suppose a sequence converges to :

Series Convergence

We say that an infinite complex series converges when the sequence of its partial sums converges:

If a series is absolutely convergent it is convergent. If converges absolutely then:

Integral Test

If is a continuous, positive, decreasing function on then:

Ratio Test

The series converges absolutely if:

Comparison Test

If and is convergent than is convergent.

Root Test

The series converges absolutely if:

Power Series

A power series about is written as:

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Radius of Convergence

The series converges absolutely whenever , where can be found by:

An alternate method to find R is to note that it is equal to the distance from the point being expanded

about ( ) to the nearest singularity.

Roots of Unity

Solve by writing in polar form:

Hence:

Thus we have:

3. Complex Limits and Derivatives

Limits from First Principles

The limit of as approaches is written as:

The limit exists if:

For a complex limit to exist, must approach the same value from every direction on the complex plane.

Complex and Real Limits

If we have , then:

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Continuity

A function is said to be continuous if:

Definition of Derivative

The derivative of a function is defined as:

Derivative of Conjugate

Let and

Along the real line so we have:

Along the imaginary line so:

The limits do not agree, so the limit does not exist.

4. Analytic Functions and Power Series

Analytic Functions

A function is analytic at a point if the derivative exists at that point, and also at every point in some

neighbourhood around the point. A function that is analytic everywhere in C is called an entire function.

An analytic function is expressible as a power series, meaning that it can be written:

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Uniform Convergence

A series is said to be uniformly convergent if for all and all values of , there is a single

such that:

This can also be written as:

In other words, should approach zero as regardless of the value of .

Uniform Convergence and Continuity

If the sequence converges uniformly to and is continuous for each , then is also

continuous. In the case of infinite series this implies:

Suppose we have converging uniformly:

Suppose we also have continuous at :

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We can thus show that must be continuous at :

Weierstrass M-Test

If and converges (no question of uniformity here, because its already

independent of z), then converges uniformly on D.

Suppose we have a series with radius of convergence . Consider now . If

then:

And we know that this latter series converges because . Thus we have shown that the series

converges uniformly.

Uniform Convergence of Power Series

A power series with radius of convergence converges uniformly for all .

Uniform Convergence Theorem

A sequence of analytic functions which converges uniformly to for all (where is

simply connected and compact) is analytic with . An important application of this is that

power series are differentiable inside their radius of convergence.

Singular Points

A singular point is a point at which fails to be analytic. Singularities are said to be isolated if they

are enclosed by an open neighbourhood continaining no other singular point.

Removable singularities: exists, so easy to make the function analytic again

Poles: can write

for other pole

Branch points: multi-valued functions like and

have a branch point at

Essential singularity: a singularity that is not removable, a pole, or a branch point. is neither

bounded nor goes to infinity (like do poles) at such singularities, but actually take on every

complex value in every neighbourhood of z=a.

5. Functions with Branches

Branch Cuts

For a multi-valued function, a branch point is a point such that the function is discontinuous when going

around an arbitrarily small circuit around this point. A branch cut is a curve (with ends possibly open,

closed, or half-open) in the complex plane across which an analytic multivalued function is

discontinuous.

To find the branch cut, write all complex powers in terms of logarithms as shown below.

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Logarithm Function

The complex logarithm is a many-valued function, defined as:

To obtain a single-valued function, it is usual to work with the principal value of the logarithm, with

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Fractional Powers

Square Root Function

The complex square root is a two-valued function:

6. Harmonic Functions

Cauchy-Riemann Equations

For a function to be differentiable, it must satisfy the Cauchy-Riemann

equations:

Cauchy-Riemann Theorem

If defined on an open region R with and being in and

satisfying the Cauchy-Riemann equations, then is analytic in .

Harmonic Functions

A real function of two variables that is and satisfies the laplace equation is said to be harmonic:

If is analytic in an open domain, then and are harmonic in

that domain.

For every harmonic , there exists a harmonic conjugate such that

is analytic. It can be found by solving the Cauchy-Riemann equations with known.

Level Curves

The level curves of a harmonic function are orthogonal to one another. Consider:

Now differentiating:

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Thus we find orthogonality:

7. Complex Integration

Smooth Curves

A smooth curve is non-self-intersecting, and is closed if the value at its starting and ending points is the

same. A contour is a finite sequence of smooth curves, not necessarily closed.

Jordan Theorem

A simple closed curve Γ divides the plane C into two open regions having the curve as their common

boundary.

Contour Integrals

A contour integral is an integral which takes a smooth curve in the complex plane as its domain of

integration, rather than merely the real axis. Contour integration is defined as:

ML Bound

The value of a contour integral is always less than or equal to the maximum value of the function over

that contour, times the length of the contour:

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Green's Theorem

Cauchy's Theorem

If is analytic on a simply-connected open domain then for any simple closed contour :

Cauchy Integral Formula

If is a simple closed contour and is analytic in a simply connected open domain then:

This is a special case of the more general formula:

Existence of Anti-Derivative

If we define, for simple contour and analytic function :

By path independence we can write this in terms of a simple direct line :

We now have:

Again by path independence we can rewrite this as:

Which by the theorem of anti-derivatives for straight line contours is equal to . Thus:

Taylor's Theorem

If is analytic in the disk then the Taylor series converges uniformly to for in disk.

With as

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8. Residue Calculus

Laurent Series

The Laurent series of a complex function f(z) is a representation of that function as a power series which

includes terms of negative degree. It may be used to express complex functions in cases where a Taylor

series expansion cannot be applied. Laurent series are defined in an annulus .

Two methods for finding laurent series:

Use partial fractions to decompose into terms directly

Separate out into known series (e.g. take out factors)

There is a relationship between the type of singularity and the number of negative power terms in the

laurent series:

Removable singularity: no negative power terms

Pole of order m: has m negative power terms

Essential singularity: infinitely many negative power terms

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Residues

If has an isolated singularity at , then the coefficient of the term

in the laurent series

for about is called the residue of at .

Limit Formula for Residues

IF has a pole of order at then:

For simple poles this simplifies to:

Note that it is often useful to use complex L'Hopital in evaluating this limit (if in numerator

and denominator).

Residue Theorem

If is a positively oriented simple closed contour and is analytic on and inside except at isolated

singularities, then:

Trigonometric Integrals

To solve an integral of the form:

Let , so

and we have:

Thus we have:

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By the residue theorem:

Using the limit formula:

Hence we find:

Integral of Rational Function over R

Consider the integral:

We have analytic on the upper half plane except for isolated points, and we also have the degree

of the denominator two or more greater than that of the numerator, so we can use the result:

Thus we have:

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Thus we have:

Jordan's Lemma

If converges uniformly to zero on as then for :

This allows us to compute integrals over the real axis in much the same was as we can do for polynomial

functions, as the 'top arc' goes to zero, so we have for so-called Fourier integrals:

Indented Contours

Things become slightly more complicated when the poles appear on the real line, as in:

We know that around the outside loop is zero by Jordan's lemma, since:

We also know that the full contour integral evaluates to zero by Cauchy's theorem, thus we have:

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Parameterise:

Removing the negative sign to evaluate in a counter-clockwise direction:

9. Consequences of Cauchy's Theorem

Gauss Mean Value Theorem

If is analytic inside and on a circle given by , then the mean of on is .

Let (analytic on a circle given by ), and so , so using the Cauchy

integral formula:

Which is clearly the average value on the circle.

Maximum Modulus Principle

If is analytic in and on a simple closed contour , then the maximum value of occurs only on

, unless is constant.

Liouville's Theorem

If is entire and bounded in , then is constant.

To prove, use the Cauchy Integral Formula:

Let (analytic on a circle given by ), and so :

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For the case :

If is entire it is analytic on an infinitely large circle, and so this inequality must apply in the limit:

Which means the function must be constant.

Fundamental Theorem of Algebra

Every polynomial of degree has exactly roots in . Suppose that had no roots. Then there

is no value of for which , and hence we can therefore say that

is entire. Clearly

is also bounded (as it limits to 0 in z). Therefore by Liouville's Theorem, must be constant.

But this contradicts the assumption that . Thus we conclude that has at least one root.

10. Conformal Transformations

Conformal Maps

A conformal map is a function which preserves the angles between two intersecting curves. All analytic

functions constitute conformal maps.

Taking the limit of the difference:

Which exists since is analytic. Thus we have:

And thus both angles change by the same amount.

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Open Mapping Theorem

If is non-constant and analytic in a connected open image set on which acts, then the range of is

also an open set.

Riemann Mapping Theorem

Every simply connected open set can be mapped by a one-to-one analytic function onto the unit disc.

Since a one-to-one analytic function is invertible, it follows that any open simply-connected domain can

be mapped onto any other so long as neither is all of .

Möbius Transformations

A Möbius transformation is any non-constant rational function of the form:

And defining:

Which is conformal at every point except its pole. It can be represented by the matrix:

Where the determinant . This means that in fact only three parameters are needed:

The elementary Möbius transformations are:

1. Translation:

2. Rescaling:

3. Rotation:

4. Inversion:

All Möbius transformations can be composed of a finite sequence of elementary transformations, and

will map to itself one-to-one.

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Key Reference

Important Trigonometric Identities

Key Power Series

For :

For :

Key Trigonometric Series

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We wish to solve the real integral:

By deformation of contours:

Writing this out in full we have:

Note that neither the rational function result or Jordan's lemma are applicable directly, so we must

show manually that and go to zero.

By the ML bound:

Observe that:

Thus we have determined that:

Now examining we set :

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We are given that:

From the above results we therefore know that:

By the residue theorem, we have:

To find the residue:

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Thus we have: