Complex Analysis

Complex Analysis

Complex Plane

Why Complex Numbers? As Bombelli pondered the solutions to
x3=5x+3x^3 = 5x + 3
, he discovered it's useful to consider not just real number, but complex numbers too. The set of complex numbers,
C\mathbb{C}
, is called the Complex Plane. We can think of the complex plane, as a set of points
x+iyx + i y
where
x,yRx, y \in \mathbb{R}
. In other words, we can think of complex numbers as numbers in
R2\mathbb{R}^2
.

Polar Coordinates

We can express complex numbers (denoted usually as z or w), using polar coordinates: a length and angle. Often we write
zz
as
z=cos(θ)+isin(θ)z = \cos(\theta) + i \sin(\theta)
  • angle is often called the argument
    • a principal argument is one where
      θ\theta
      is between
      π,π\pi, -\pi
      .
e.g., Arg(1 - i) =
π4\frac{- \pi}{4}
The lenght of numbers in the complex plane, called the modulus, is computed as:
z=x2+y2\| z \| = \sqrt{ x^2 + y^2 }
The complex conjugate of a number
zz
, denoted
zˉ=xiy\bar{z} = x - i y
.
De Moivre's Formula
Since we know
(eiθ)n=einθ(e^{i\theta})^n = e^{i n \theta}
and that
eiθ=cos(θ)+isin(θ)e^{i \theta} = cos(\theta) + i \sin(\theta)
(Euler's Formula), De Moivre's Formula is the observation that:
cos(θ)+isin(θ)=cos(nθ)+isin(nθ)\cos(\theta) + i sin(\theta) = \cos(n \theta) + i \sin(n \theta)
eiθ=cosθ+isinθe^{i \theta} = \cos \theta + i \sin \theta
, because the real and imaginary parts of
eixe^{i x}
correspond to the power series of
cos\cos
and
sin\sin
.
The triangle inequality and field axioms still hold in
C\mathbb{C}

Nth Roots

An nth root of a complex number,
ww
, is a number
zz
such that
zn=wz^n = w
. An important type of nth root are the nth roots of unity, which are numbers whose nth power is 1.
primary::roots unity
There are n numbers satifying
zn=1z^n = 1
. Why?
Let's think in polar coordinates:
reinθ=1r e^{i n \theta} = 1
. Then r =1 and
nθ=2πkn \theta = 2 \pi k
for k
Z\in \mathbb{Z}
. We then have
θ=0,2π/n,4π/n,\theta = 0, 2 \pi / n, 4 \pi / n, \dots
there are n unique angles!
We can think of other nth roots as roots of unity corrected for length.

Complex Functions

A complex function is continuous in the complex plane in an analogous way as it is in the reals. Also note, the Extreme Value Theorem holds in the complex plane as it does in the reals: a continuous function on a compact set attains its min\/max.
A compelx function on an open set in
C\mathbb{C}
is holomorphic (analytic or differentiable) at a point z_0 if the dervative at z_0 exists. Note the key difference between a real and compelx derivative is that you can approach a complex point z_0 from infinitely many directions, not just left or right—it's a much stronger condition.
Nice Properites:
• sum\/product of holomorphic functions is holomorphic
• quotient, f\/g, of holomorphic functions such that g(z_0) ≠ 0, is also holomorphic
We can also think of a complex function as a function in
R2\mathbb{R}^2
where f = u(x, y) + i v(x, y), with v, u real-valued function on the real plane.
The Jacobian of a function f(x, y) is matrix of partial derivatives.

Cauchy Riemann Condition

It turns out a holomorphic function at a point satisfies the Cauchy Riemann Equations:
ux=vy\frac{\partial u}{\partial x} = \frac{\partial v }{\partial y}
and
uy=vx\frac{\partial u}{\partial y} = - \frac{\partial v }{\partial x}
use limits to show this is true with the realization the derivative should be the same regardless of the direction you approach the point with.
Furthermore, if u and v have continuous derivatives (continuously differentiable) and the Cauchy Riemann Equations hold then f is holomorphic (in the open set on which f is defined).

Power Series

[in progress]

Text Reference

Complex Analysis, by Stein and Shakarchi

Other references

A First Course In Complex Analysis, by Beck, Marchesi, Pixton, and Sabalka
Functions of One Complex Variable, by John Conway (used in Course)