complex number maths

see also:

• a brief summary of number theory
• a brief summary of mathematical principles and tools

• complex numbers were invented in the 16th century to help solve cubic equations
• z = x + yi where z = complex, x and y are real and i = sqrt(-1)
• complex numbers are used extensively:
  • to simplify signals in the time domain into their constitutive signals in the frequency domain by using Fourier and LaPlace transforms
    • this is used in almost every area of modern life - electrical and sound engineering, sound and image compression technology, radio transmission technologies, etc
  • in electrical engineering to represent the phase change component of impedance which is a complex number containing resistance and phase components - see frequency response analyzers
  • in electromagnetism where electric and magnetic fields, as well as electromagnetic waves, are modeled using complex notation for phase relationships
  • in quantum mechanics where the wave functions and probabilities are described using complex numbers
  • in fluid dynamics where they are used to describe potential flows and streamline functions in two-dimensional systems
  • in geometry and fractals where they are used to construct fractals (e.g., Mandelbrot set) and for rotations or transformations in the plane
  • for solving polynomials as every non-constant polynomial equation has a solution in the complex plane, making mathematics complete in this context
  • in cryptography were some encryption algorithms use complex number concepts
  • in graphics and simulation where image rotation, 2D/3D rendering, and visualizing data or complex functions all utilize complex numbers
  • in artificial intelligence where neural networks and data structures may be expressed via complex numbers for certain algorithms
  • in financial modeling where they are used for analyzing trends, stock prices, and economic variables
  • population dynamics and neuronal networks: Models population changes and neural activity using systems involving complex numbers
  • etc

• i = sqrt(-1) but in electrical engineering the i is generally denoted as j as I is used for current
• rectangular coordinates form:
  • z = x + yi
• polar coordinate form:
  • based on a circle in a complex plane
    • on a plot, r is the distance from origin and θ is the angle drawn from the x-axis (a unit circle obviously has radius r = 1)
    • where x axis value (the real component) is represented by r(cosθ) and the y axis by r x i x (sinθ)
    • each time you multiply by i, it rotates it anti-clockwise by 90deg such that every i⁴ rotates it a full circle of 360deg
    • if you raise i to a negative power, you rotate it clockwise such that i^-1 = 0 - 90deg = 270deg = i³
    • z = r(cosθ + i x sinθ) where θ is in degrees or radians and often called the “argument” of z

• use x=rcosθ and y=rsinθ

• use r = sqrt(x²+y²)​ and θ = arctan(y/x​​)
• the quadrant of a complex number does matter when you are converting from rectangular form to polar form:
  • the result from your calculator of arctan(y/x​​) will often be an angle in Quadrant I or IV due to the defined range of the arctan function, this needs to be corrected as follows:
    • Quadrant I (a>0, b>0): The calculator result is correct.
    • Quadrant II (a<0, b>0): Add 180∘ (or π radians) to the calculator result
    • Quadrant III (a<0, b<0): Add 180∘ (or π radians) to the calculator result
    • Quadrant IV (a>0, b<0): The calculator result is correct (or add 360∘ to get a positive angle)
  • in the context of electrical engineering, RHP typically refers to the Right-Half Plane of the complex s-plane:
    • when analyzing a circuit's behavior over a range of frequencies, to describe the circuit's transfer function, we use the complex frequency variable s = sigma+jomega (j = sqrt(-1) )
      • this transfer function, which can represent impedance, is a ratio of polynomials
      • the roots of the denominator polynomial are called poles, and the roots of the numerator polynomial are called zeros
      • plotting these poles and zeros on the complex s-plane is a key part of control systems and circuit analysis
      • the location of poles and zeros in the complex plane, particularly in the RHP, is crucial for determining a circuit's stability and dynamic behavior.
      • the complex plane is divided into a Left-Half Plane (LHP), a Right-Half Plane (RHP), and the imaginary axis (jomega)
      • Poles in the RHP (sigma0):
        • a system with a pole in the RHP is unstable. This means that a disturbance will cause the system's response to grow exponentially over time. For an impedance, this would imply an unstable or non-passive circuit, where energy is generated rather than dissipated.
      • Zeros in the RHP:
        • a zero in the RHP doesn't cause instability, but it is a sign of a non-minimum phase system. These systems are challenging to control because they introduce a phase lag, even while the gain is increasing.

• the rules differ depending on whether the numbers are in rectangular form or polar form

• assume we have two complex numbers: z₁ = a+bi and z₂ = c+di
• addition of z₁ + z₂:
  • z₁+ z₂ = (a+c)+(b+d)i
• multiplication of z1 x z2 is similar to multiplying two binomials using the FOIL (First, Outer, Inner, Last) method:
  • z₁ x z₂ = (a+bi)(c+di) = ac+adi+bci+bdi² = (ac−bd)+(ad+bc)i

• assume we have two complex numbers: z₁ = r₁(cos theta₁ + i x sin theta₁) and z₂ = r₂(cos theta₂ + i x sin theta₂)
• addition of z₁ + z₂:
  • you must first convert them to rectangular form, add them, and then convert the result back to polar form
• multiplication of z₁ x z₂ is done by multiplying the magnitudes and adding the angles:
  • z₁ x z₂ = (r₁ x r₂) x (cos (theta₁ + theta₂) + i x sin(theta₁+ theta₂) )
    • NB. the modulus or r₁ and modulus of r₂ are usually used in this calculation where modulus of r₁ = sqrt(r₁²)

• NB. remember that from a brief summary of logarithms and exponential functions:
  • e^x is NOT defined as being equal to e multiplied by itself x times as is the case in usual terminology of 2^x although for x as a Real Number, it does equate to this, but not if x is a complex number
    • INSTEAD it refers to a different infinite polynomial function, often called exp(x) where x can be a real number, complex number, matrix or a quantum operator
      • exp(x) = 1 + x + (x)²/2 + (x)³/6 + ….+ (x)^N/N! = sum( (x)^N/N! ) for N = 0 to however much you want to get to, eg. 100, remembering that for N= 0, 0! = 1 which is the 1st term in our equation, and N=1 gives x, the 2nd term in our equation.
      • ie. exp(0) = e⁰ = 1
      • ie. exp(1) = e¹ = e = 1 + 1 + (1)²/2 + (1)³/6 + ….+ (1)^N/N! = 2.71828…
  • exp(a+b) = e^(a+b) = e^a x e^b
    • this means that when x is a Real number, e^x DOES equate to e multiplied by itself x times as:
      • eg. if x = 5 then exp(5) = e⁵ = e^(1+1+1+1+1) = e¹ x e¹ x e¹ x e¹ x e¹ = e x e x e x e x e
  • d/dt e^nt = n x e^nt
    • thus d/dt e^it = i x e^it and multiplying it by i rotates it ant-clockwise 90deg on the complex plane
• e^iθ = exp(iθ) = 1 + iθ + (iθ)²/2 + (iθ)³/6 + …. + (x)^N/N!
  • ie. if each subsequent component of this polynomial is represented as a vector adding to the previous position, the equation forms a spiral of vectors on a complex plane with exp(iθ) being on the unit circle in the complex plane
• Euler's formula: e^iθ = cos(θ) + i x sin(θ)
  • ie. the complex exponential e^iθ is a point on the unit circle in the complex plane
• Euler's identity equation, for when θ = π radians it simplifies to:
  • e^iπ = -1
    • NB. remembering, π radians is 180deg and cosine of π radians is -1 and sine of π radians is 0
    • hence, also, e^i2π = 1 as 2π will rotate it 360deg and back to where you started in the complex plane and cosine of 2π radians is 1 and sine of 2π radians is 0
    • see https://www.youtube.com/watch?v=v0YEaeIClKY 3Blue1Brown's explanation
    • https://www.youtube.com/watch?v=mvmuCPvRoWQ&pp=0gcJCcYJAYcqIYzv 3Blue1Brown's explanation in Group Theory

• log_e(cosθ + isinθ) = iθ