Complex Analysis in Practice
Unlock the visual and computational power of complex numbers, from conformal mappings to contour integration. Master the Cauchy-Riemann equations and use the residue theorem to effortlessly evaluate notoriously difficult real integrals.
Like Duolingo, but for Complex Analysis in Practice. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
26 levels across 4 sections, about 52 minutes end to end, roughly 10 days at five minutes a day. It moves through The Geometry of Complex Numbers and Mappings; Derivatives and the Rigidity of Analyticity; Paths and Contour Integration; and Singularities, Residues, and Real Integrals. It starts from scratch.
Free forever · No credit card · iPhone & Android
Key ideas in Complex Analysis in Practice
- A complex number a + bi can be plotted as a static point or position vector (a, b) in the complex plane
- Adding a fixed complex number shifts every point across the plane by that vector
- Multiplying by an imaginary unit like i acts as a rotation of the entire plane
- When multiplying two complex numbers, their distances from the origin (lengths) multiply together
- Their angles measured from the positive real axis add together
- Complex multiplication is fundamentally a combined stretch and rotation
- In exponential form e^(i·θ), taking a power multiplies the exponent: (e^(i·θ))^n = e^(i·nθ)
- Raising a complex number on the unit circle to a power simply repeats its rotation n times
- An angle of 2π represents a full 360° rotation back to 1 on the real axis
- The nth roots of unity are numbers that equal 1 when raised to the nth power
- Because their angles must multiply to multiples of 360°, they space themselves at equal steps of 360°/n around the unit circle
- Solving z^n = 1 geometrically carves the circle into a regular n-sided polygon with one vertex at 1
- The complex conjugate of a + bi is a - bi, which mirrors the point vertically across the real axis
- Multiplying any complex number by its conjugate gives (a + bi)(a - bi) = a² + b², which is always a real number
- Clearing the imaginary term from a denominator turns division into simple division by a real scalar
- A complex number has two real dimensions (real and imaginary parts)
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Complex Analysis in Practice the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
Picture a swarm of dots on the grid. If you add 3 + 2i to every single dot, what happens to them?
Get it right to open this lesson and 25 more in the app.
How to learn Complex Analysis in Practice, step by step
Go in order. 26 short levels across 4 sections, about 52 minutes in all, five minutes a day.
- 1
The Geometry of Complex Numbers and Mappings
- Calculating with Geometry: Addition, Scaling, and Rotation
- Mapping the Plane: How Complex Functions Transform Space
- 2
Derivatives and the Rigidity of Analyticity
- Detecting Differentiability with Cauchy-Riemann Equations
- Reconstructing Fields with Harmonic Conjugates
- 3
Paths and Contour Integration
- Evaluating Integrals Along Parametric Paths
- Exploiting Cauchy’s Integral Theorem and Formula
- 4
Singularities, Residues, and Real Integrals
- Classifying Singularities with Laurent Series
- Computing Hard Real Integrals via the Residue Theorem
4 sections · 8 units · 26 levels. Built to play, not to enroll.
Questions about Complex Analysis in Practice
How do I learn Complex Analysis in Practice?
Go in order: start with The Geometry of Complex Numbers and Mappings, then Derivatives and the Rigidity of Analyticity, then Paths and Contour Integration, then Singularities, Residues, and Real Integrals. On Tomo that's 26 short levels, about 52 minutes in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for Complex Analysis in Practice?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Complex Analysis in Practice is one of its free courses, and you can type any other topic to get a course of your own.
Is the Complex Analysis in Practice course free?
Yes. Complex Analysis in Practice is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn Complex Analysis in Practice?
26 levels across 4 sections, about 52 minutes of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start Complex Analysis in Practice?
No. It's pitched at a beginner level, so it starts where you are and builds up. If you can tap a phone, you can start.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Complex Analysis in Practice is taught in the Storyteller style: every lesson is a story. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Complex Analysis in Practice today.
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