A complex number is a number that is written as where is the imaginary unit defined as and . The set of all complex numbers is written as , it is an extension if The Real Numbers
Complex numbers have lots of applications as they are helpful when describing waves due to their connection with the Trigonometric Functions. They are also used in physics due to their ability to describe rotations and oscillations in 2 Dimensions neatly.
The Complex Conjugate
For , is always real as the imaginary parts cancel so is also always real as it’s a difference of two squares
Forms of Complex Numbers
Regular
Basic operations
Modulus Argument Form
Complex numbers can also be written with Trigonometric Functions in the form where and .
The modulus of a complex number is written as and it is is defined as where . is geometrically represented as the distance from the origin to on the complex plane.
The argumentwritten as is the angle such that - it is given in radians typically given in the range (This is called the principal argument) This angle represents the angle rotation from the -axis
Basic Operations
Multiplication
Division
Exponential Form
Every complex number can be written in the form where . Complex exponentiation is defined using the Taylor series of as it only contains integer powers of which can be extended for complex inputs .
Intuition is the unique function such that with this relation means that the “velocity” of is proportional to itself.
Considering we get that’s it derivative is meaning it’s velocity is a rotation of itself. So starting at it’s velocity is always tangent to it which describes circular motion so it can be described by travelling around a unit circle and hence it can described by trig functions.
Properties
We can also split the power into multiplication similar to real exponentiation as .
Proof
Using Taylor series we get as well as . We can prove they are equivalent by comparing coefficients.
For in we can only get these terms from the expansion of which itself also has a coefficient of . The coefficient from the expansion is - the comes from the . Putting them together we get .
With we can only get by picking the term and the term in the expansion. The has a coefficient of and the has a coefficient of which when multiplied gives the same coefficient of since every term has the same coefficient they are equal
Complex Logarithm
Using this exponential definition we can extend the domain of to . Using the defining property that .
Since we have extended to we can simply define and since every complex number can be written in this form except we can extend it. Writing a complex number in the form we get and using the fact that we get the full definition that
However since is not bijective over a proper inverse of can’t exist. This comes down to the fact that so is not unique. Typically this is dealt with by simply restricting to however other branches are valid it is just dependant on context.
We can also deal with where by simply using the change of base formula to use .
General Complex Exponentiation
Given two complex numbers we can compute as .
De Moivre’s Theorem
Proved simply by using induction and using the rules how adds/subtracts and multiplies/divides.
It can be used to find Trigonometric Identities by applying De Moivre’s Theorem and the binomial expansion of to express in terms of powers of and same for
You can also use the following identities
to find other identities by binomially expanding the function in terms of and exponentiating the trig function. These identities themselves can be proved using De Moivre’s Theorem.
Roots of Complex Numbers
To Solve,
Assume has the form so by De Moivre’s Theorem
Compare the modulus to solve for
Then plug in and solve for
Roots of Unity
The roots of unity are the solutions to ,
If is a positive integer there is an th root of unity s.t are the roots of unity
The roots of unity sum to 0 and form the vertices of a regular -gon
If is one root of the equation then the roots of are given by