Cauchy–Riemann Equations
Conditions for complex differentiability.
Key idea
Conditions for complex differentiability.
In depth
For f = u + iv to be complex differentiable, u_x = v_y and u_y = −v_x (plus continuity of partials suffices). They express that f respects the complex multiplication structure, so it acts locally as a rotation and scaling.
Worked example
f = z²: u = x² − y², v = 2xy; u_x = 2x = v_y, u_y = −2y = −v_x. ✓
Formula
Real-world uses
- Checking whether a two-dimensional physical model, such as airflow, is consistent
Common mistake
Verifying only one of the two equations.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cauchy–Riemann Equations?
Conditions for complex differentiability. It is a topic at the expert level in Analysis & Topology.
What is the formula for Cauchy–Riemann Equations?
The key formula is uₓ = vᵧ, uᵧ = −vₓ.
Where is Cauchy–Riemann Equations used in real life?
Real-world examples: Checking whether a two-dimensional physical model, such as airflow, is consistent.
What should I learn before Cauchy–Riemann Equations?
Make sure you are comfortable with Complex Differentiability, the previous topic in Analysis & Topology.
What is a common mistake with Cauchy–Riemann Equations?
Verifying only one of the two equations.
How can I practise Cauchy–Riemann Equations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cauchy–Riemann Equations?
Move on to Harmonic Functions next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.