Complex Analysis

  • Notes
  • Videos & Interactives
  • Questions
  • Exam Questions
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Complex Numbers 
Analytic Functions 
Zeroes and Singularities 
Elementary Functions 
​Complex Integration 
Sequences and Series 
​Series Representation of Analytic Functions 
Residue Theory 


​​Complex Numbers
​Complex Numbers
Arithmetic of Complex Numbers
Arithmetic of Complex Numbers with Dienes Blocks
Arguments and Polar Form of Complex Numbers
​Multiplication and Division in Polar Form
​Proof of Basic Identities involving Modulus and Argument

​​Relations of the Complex Plane​
Loci Definitions using Arguments on the Complex Plane
More Loci Definitions using Arguments on the Complex Plane 
Conjugates and Relations on the Complex Plane 
Distance Definitions of Lines, Circles, and Ellipses 
Distance Definitions of Hyperbolas and Conic Sections 
Regions of the Complex Plane

Powers of Complex Numbers
​Roots of Complex Numbers
Exponential Form of Complex Numbers 

​Factorising Polynomials over C
​Solving Polynomial Equations over C
The Conjugate Root Theorem​
Intersections on the Complex Plane Algebraically 

Technology​
​Complex Numbers Technology [TI-nSpire]
​Complex Numbers Technology [Casio]
​Complex Locus Plotter (GeoGebra Interactive)

what if we have a negative number inside of a logarithm? (blackpenredpen Video)​
Analytic Continuation and the Zeta Function (zetamath Video)

How to Solve 2D Equations using Color (3Brown1Blue Video)
2-D Color Solver (GeoGebra Interactive)
Zeroes (Berkley Interactive)
Zeroes and Singularities (Berkley Interactive)
Zeroes and Singularities Explanation (Kyle Miller Article)

​Complex Integration and Finding Zeros of the Zeta Function (zetamath Video)

Maclaurin Series Approximations (GeoGebra Interactive)
​
you didn't use complex numbers to integrate 1/(x^2+1) in calculus 2 (blackpenredpen Video)

Stabbing Man: z →z² Transformation (GeoGebra Interactive)
Complex Mapping Visualiser (Mabotkin Interactive)

Fluid Flow with Complex Functions
Visualising the Riemann Zeta Function and Analytic Continuation (3Brown1Blue Video)
Pi Hiding in Prime Regularities ​(3Brown1Blue Video)

Lockdown Math
Complex Polynomial Roots (Leo C. Stein Interactive)

​Who cares about complex numbers? (Eddie Woo Video)

The Most Beautiful Identity
Imaginary Numbers are Real
This Problem Broke Math (and led to quantum physics) (Veritasium Video)
Visualising Quaternions - An Explorable Video Series (3Brown1Blue Videos & Article)
Complex Analysis (Juan Carlos Ponce Campuzano Interactive Article)

500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle? (Mathologer Video)
Newton's Fractal (which Newton knew nothing about) (3Blue1Brown Video)
Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”) (3Blue1Brown Video)

PHANTOM GRAPHS (Philip Lloyd Notes)

What does it mean to take a complex derivative? (visually explained) (vcubingx Video)

What does the Laplace Transform really tell us? A visual explanation (Zach Star Video)
The Beauty behind Complex Numbers (Zach Star Video)
How do complex numbers actually apply to control systems? (Zach Star Video)
Fourier and Laplace Transforms Explained Geometrically (Zach Star Video)
A Calculus 2 Mystery Solved with Imaginary Numbers (Zach Star Video)
The Intuition and Implications of the Complex Derivative (Zach Star Video)

An Interactive Introduction to Fourier Transforms (Jezzamon Interactive Article)


Rational Functions of a Real Variable
Multiplying rational expressions (Khan Academy Video)
Dividing rational expressions (Khan Academy Video)

Partial Fractions
​Quadratic Factors
Partial Fractions: Repeated Linear Factors - ​Technique for Breaking Down into Partial Fractions (Eddie Woo Video)
​

Complex Numbers
A Visual, Intuitive Guide to Imaginary Numbers (Better Explained Article)
​This Problem Broke Math (and led to quantum physics) (Veritasium Video)
Dienes Blocks (Maths Bot Interactive)​

​Introduction to Complex Numbers
​Complex Arithmetic
Complex Arithmetic
​Complex Numbers as Points
Complex Locus Plotter (GeoGebra Interactive)

Graphs in the Complex Plane
Further Graphs of the Complex Plane
Graphs in the Complex Plane
Semi Circles on Argand Diagrams
Algebraic Proof for Opposing Rays  ​
Arcs on the Complex Plane
​Graphing in the Complex Plane: Worked examples (Eddie Woo Video)
Sketching Subsets of the complex Plane (Joel Speranza Math Video)
Sketching Sets Defined on the Complex Plane (Maths Stats UNSW Video) [7:39 - 13:48] ​

​Square Roots of Complex Numbers
Introducing Polar Form
Complex Numbers - Mod-Arg Form
Multiplying Complex Numbers in Mod-Arg Form
Using Inverse Tan to Find Arguments?
Working in Polar Form 
Exact value of cos(π÷12) (Eddie Woo Video)
Complex Numbers: A Very Fast Recap of i, Operations, Conjugates, Argand Diagrams & Modulus (Joel Speranza Math Video)
Complex Numbers: Polar Form: Fast Recap including Operations on Polar Form (Joel Speranza Math Video)
​Relationships Between Moduli & Arguments in Products of Complex Numbers (Eddie Woo Video)
​Understanding Complex Quotients & Conjugates in Mod-Arg Form (Eddie Woo Video)

Quotient of Complex Numbers
Argument of the Complex Conjugate (Eddie Woo Video)
Working with Moduli and Arguments (Proof Question) (Eddie Woo Video)

​The Triangle Inequalities ​
​De Moivre's Theorem (Eddie Woo Video)
​Using De Moivre's Theorem and Binomial Expansions (Eddie Woo Video)

Proving de Moivre's Theorem
Using de Moivre's Theorem - Example Question
An Interactive Approach to Finding nth Roots (Sine of the Times Interactive)
Factorising & Solving quadratics over C (Joel Speranza Math Video)

Equations with Complex Solutions
Relation to square roots (Eddie Woo Video)
Solving & factorising (Eddie Woo Video)

​Linear Factorisation of Polynomials
Complex Conjugate Root Theorem
Complex Conjugate Root Theorem / Roots of Unity
​Complex Conjugate Root Theorem
Conjugate root theorem: What it is and Proof (Joel Speranza Math Video)
Factorising Polynomials over C worked examples (Joel Speranza Math Video)
Complex Polynomial Roots (Leo C. Stein Interactive)
Roots of Unity (Joel Speranza Math Video)

Roots of Unity
Nth Roots of a ℂ Number 
Calculating Roots of a complex number using De Moivre's Theorem (Joel Speranza Math Video)

​Who cares about complex numbers? (Eddie Woo Video)
Complex Number Fundamentals (3Blue1Brown Video)
Discovering Euler's Trigonometric Formula (James Tanton Video)

The Most Beautiful Identity (Optional)
What does an imaginary power mean? (Eddie Woo Video) [Optional]

Informal Proof of Euler's Formula (Optional)
Proving Euler's Formula (Optional) ​
Trigonometric Expansions from Complex Numbers (Optional)
​Imaginary Numbers are Real
PHANTOM GRAPHS (Philip Lloyd Notes)
​Real Part and Imaginary Part (mathworksheets4kids)
Plotting Complex Numbers (mathworksheets4kids)
Complex Numbers: Complex Conjugate (Variation Theory)
​Complex Numbers: Adding and Subtracting (Variation Theory)
Adding and Subtracting Complex Numbers (mathworksheets4kids)
Complex Numbers: Multiplication (Variation Theory)
Multiplying Complex Numbers (mathworksheets4kids)
Complex Numbers: Division (Variation Theory)
Rationalise the Denominator (mathworksheets4kids)
Simplifying Complex Numbers (mathworksheets4kids)
Powers of i (mathworksheets4kids)
Absolute Value of Complex Numbers (mathworksheets4kids)
Absolute Value and Argument of Complex Numbers (mathworksheets4kids)​
Complex Numbers Polar Form (Starting Points Maths)
Using De Moivre's Theorem to Evaluate Powers of Complex Numbers (Variation Theory)
Finding the nth Root of Complex Numbers (Variation Theory)
​Complex Numbers: Polynomials from Roots (Variation Theory)
​Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 

Complex Numbers (2016-2022 Study Design Questions)
Complex Numbers (2000-2015 Study Design Questions)
Relations of the Complex Plane (2016-2022 Study Design Questions)
Relations of the Complex Plane (2000-2015 Study Design Questions)
​Regions on the Complex Plane (2000-2015 Study Design)
​Polynomials over C (2016-2022 Study Design Questions)
​Polynomials over C (2000-2015 ​Study Design Questions)