1_complex_numbers.md



title: Complex Numbers


1. Complex Numbers
The lecture on complex numbers consists of three parts, each with their own video:

1.1. Definition and basic operations
1.2. Complex functions
1.3. Differentiation and integration

Total video length: 38 minutes and 53 seconds

1.1 Definition and basic operations


Complex numbers
!!! info "Definition I"
Complex numbers are numbers of the form z = a + b {\rm i}.
Here \rm i is the square root of -1: {\rm i} = \sqrt{-1},
or equivalently: {\rm i}^2 = -1.
Usual operations on numbers have their natural extension for complex
numbers, as we shall see below.
Some useful definitions:
!!! info "Definition II"
For a complex number z = a + b {{\rm i}}, a is called the real part, and b the imaginary part.
!!! info "Complex conjugate"
The complex conjugate z^* of z = a + b {{\rm i}} is defined as
z^* = a - b{{\rm i}},
i.e., taking the complex conjugate means flipping the sign of the imaginary part.

Addition
!!! info "Addition"
For two complex numbers, z_1 = a_1 + b_1 {{\rm i}} and z_2 = a_2 + b_2 {{\rm i}},
the sum w = z_1 + z_2 is given as
w = w_1 + w_2 {{\rm i}}= (a_1 + a_2) + (b_1 + b_2) {{\rm i}}
where the parentheses in the rightmost expression have been added to group the real and the imaginary part. A consequence of this definition is that the sum of a complex number and its complex conjugate is real:
z + z^* = a + b {{\rm i}}+ a - b {{\rm i}}= 2a, i.e., this results in twice the real part of z.
Similarly, subtracting z^* from z yields z - z^* = a + b {{\rm i}} - a + b {{\rm i}}= 2b{\rm i}, i.e., twice the imaginary part of z (times \rm i).

Multiplication
!!! info "Multiplication"
For the same two complex numbers z_1 and z_2 as above, their product is calculated as
w = z_1 z_2 = (a_1 + b_1 {{\rm i}}) (a_2 + b_2 {{\rm i}}) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) {{\rm i}},
where the parentheses have again beèn used to indicate the real and imaginary parts.
A consequence of this definition is that the product of a complex number
z = a + b {{\rm i}} with its conjugate is real:
z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2.
The square root of this number is called the norm |z| of z:
|z| = \sqrt{z z^*} = \sqrt{a^2 + b^2}.

Division
The quotient z_1/z_2 of two complex numbers z_1 and z_2 defined above can be evaluated by multiplying the numerator and denominator by the complex conjugate of z_2:
!!! info "Division"
\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.
Try this yourself!
!!! check "Example:"
\begin{align} 
\frac{1 + 2{\rm i}}{1 - 2{\rm i}} &= \frac{(1 + 2{\rm i})(1 + 2{\rm i})}{1^2 + 2^2} = \frac{1+4{\rm i} -4}{5}\\
& = -\frac{3}{5} + {\rm i} \frac{4}{5}
\end{align}

Visualization: the complex plane
Complex numbers can be rendered on a two-dimensional (2D) plane, the
complex plane. This plane is spanned by two unit vectors, one
horizontal representing the real number 1 and the vertical
unit vector representing {\rm i}.

  
  The norm of $z$ is the length of its vector spanned in the complex plane.


Addition in the complex plane
Adding two numbers in the complex plane corresponds to adding their
respective horizontal and vertical components:

  
  The sum of two complex numbers is found as the diagonal of a parallelogram spanned by the vectors of those two numbers.


1.2. Complex functions

Real functions can (most of the times) be written in terms of a Taylor series expanded at a point x_{0}:
f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (x-x_{0})^{n}
We can write something similar for complex functions by replacing the real variable x with its complex counterpart z:
f(z) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (z-x_{0})^{n}
For this course, the most important function is the complex exponential function, at which we will have a closer look below.

The complex exponential function
The complex exponential is used extremely often.
It occurs in Fourier transforms and it is very convenient for doing calculations involving cosines and sines.
It also makes many common operations on complex number a lot easier to perform.
!!! info "The exponential function and Euler identity"
The exponential function f(z) = \exp(z) = e^z is defined as:
\exp(z) = e^{x + {\rm i}y} = e^{x} e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).
The last expression is called the Euler identity.
!!! note "Exercise"
Check that this function obeys
\exp(z_1) \exp(z_2) = \exp(z_1 + z_2).
You will need sum and difference formulas of cosine and sine.

The polar form
A complex number z can be represented by two real numbers, a and b, which correspond to the real and imaginary part of the complex number.
Another representation of z is a vector in the complex plane with a horizontal component that corresponds to the real part of z and a vertical component that corresponds to the imaginary part of z.
It is also possible to characterize that vector by its length and direction, where the latter can be represented by the
angle that the vector makes with the horizontal axis:

  
  The angle with the horizontal axis is denoted by $\varphi$ 
  like in the case of conventional polar coordinates, 
  but in the context of complex numbers, this angle is called as the argument.

!!! info "Polar form of complex numbers"
A complex number can be represented either by its real and imaginary part
corresponding to the Cartesian coordinates in the complex plane,
or by its norm and its argument corresponding to polar coordinates.
The norm is the length of the vector, and the argument is the angle it makes with the horizontal axis.
We can conclude that for a complex number z = a + b {\rm i}, its real and imaginary parts
can be expressed  in polar coordinates as a = |z| \cos\varphi b = |z| \sin\varphi
!!! info "Inverse equations"
The inverse equations are |z| = \sqrt{a^2 + b^2}
\varphi = \arctan(b/a) for a>0.
In general:
\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
\pi + \arctan(b/a) & {\rm for ~} a<0 {\rm ~ and ~} b>0;\\
-\pi + \arctan(b/a) &{\rm for ~} a<0 {\rm ~ and ~} b<0. \end{cases}
It turns out that by using the magnitude |z| and phase \varphi, we can write any complex number as
z = |z| e^{{\rm i} \varphi}
By increasing \varphi by 2 \pi, we make a full circle around the origin and reach the same point on the complex plane. In other words, by adding 2 \pi to the argument of z, we get the same complex number z!
As a result, the argument \varphi is defined up to 2 \pi, and we are free to make any choice we like, such as in the examples in the figure below:

  
   $-\pi < \varphi < \pi$ (left) and (right) $-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2}$ 

Some useful values of the complex exponential to know by heart are:
!!! tip "Useful identities:"
e^{2{\rm i } \pi} = 1
e^{{\rm i} \pi} = -1 
e^{{\rm i} \pi/2} = {\rm i}
From the first expression, it also follows that
e^{{\rm i} (y + 2\pi n)} = e^{{\rm i}y} {\rm ~ for ~} n \in \mathbb{Z}
As a result, y is only defined up to 2\pi.
Furthermore, we can define the sine and cosine in terms of complex exponentials:
!!! info "Complex sine and cosine"
\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}
\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}
Most operations on complex numbers become easier when complex numbers are converted to their polar form using the complex exponential.
Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by substituting the real variable x with the complex variable z in its polar form:
!!! info "Examples of some complex functions stated using polar form"
z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}
\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} 
\log(z) = log \left(r e^{{\rm i} \varphi}\right) = log(r) + {\rm i} \varphi
z_{1}z_{2} = r_{1} e^{{\rm i} \varphi_{1}} r_{2} e^{{\rm i} \varphi_{2}} = r_{1} r_{2} e^{{\rm i} (\varphi_{1} + \varphi_{2})}