Complex Mathematics
cmath is a class used for calculations using complex numbers. This module is necessary when calculating Fourier transforms, but it can be used for a lot of other purposes as well.
Required by another script
This module must have the same Parent as the Fourier module.
Creating a new complex number
ComplexNumber cmath.new(float real?, float imag?)
Creates a new Complex Number object. real is the real component and imag is the imaginary component.
ComplexNumber cmath.fromDict(dict {float Real?, float Imag?})
Creates a new Complex Number object using a dictionary instead of two separate arguments.
Properties
float Real
The real component of the Complex Number.
float Imag
The imaginary component of the Complex Number.
Methods
float, float polar()
Returns the polar representation of a Complex Number. First number is the distance from origin (magnitude) and the second is the phase angle in radians. Magnitude is calculated using the abs method. Phase angle is calculated with $$ \theta=\mathrm{atan2}\left(b{,}\ a\right) $$
ComplexNumber exp()
Returns e (approximately 2.7182818) raised to the power of a Complex Number. Uses the formula $$ e^z=e^a\cos b+ie^a\sin b{,}\ \mathrm{where}\ z=a+bi\ \mathrm{and}\ a{,}\ b\in\mathbb{R} $$
float abs()
Returns the absolute value (distance from origin) of a Complex Number with $$ \left|z\right|=\sqrt{a^2+b^2}{,}\ \mathrm{where}\ z=a+bi\ \mathrm{and}\ a{,}\ b\in\mathbb{R} $$
Important
When performing comparisons <, >, <= and >= with complex numbers, the absolute value is used as there is no universally accepted way to compare complex numbers with each other. When using the == comparison, the real and complex components are compared individually.
Mathematical operations
This section lists all mathematical operations supported by Complex Numbers. In this section \(z\) and \(w\) can both be either complex or real numbers. When necessary, real number inputs are automatically converted to complex numbers. In the following examples, \(z=a+bi\) and \(w=c+di\), where \(a{,}\ b{,}\ c{,}\ d\in \mathbb{R}\).
Addition and subtraction
To perform addition or subtraction with complex numbers, you can simply use z+w and z-w in your code. The formula used for this is
$$
z+w=a+c+(b+d)i
$$
Multiplication
Multiplication is also supported using the standard * operator. This module performs complex multiplication as follows:
$$
z\cdot w=\left(ac-bd\right)+\left(ad+bc\right)i
$$
Division
Division can be done using the / operator for any \(z\) and \(w\). Division uses the following formula:
$$
\frac{z}{w}=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i
$$
Exponentiation
Exponentiation can be done using the ^ operator for any \(z\) and \(w\). The result is calculated as follows:
$$
z^w=e^{\log r\left(c+di\right)+\theta i\left(c+di\right)}
$$
where \(r\) and \(\theta\) are the magnitude and phase angle of the polar representation of the base.