Mathematics · Advanced university
Holomorphic functions and Cauchy-Riemann equations · revision notes
A derivative must have the same limit along every complex direction.
Key ideas and conditions
Complex differentiability
A derivative must have the same limit along every complex direction.
Cauchy-Riemann
For f=u+iv, differentiability implies u_x=v_y and u_y=−v_x when the partials exist.
Sufficient hypotheses
Continuous partial derivatives satisfying the equations on a neighborhood give holomorphicity there.
Not a point test
Equations at one isolated point alone do not prove holomorphicity throughout a neighborhood.
Explain without looking
- Why is real differentiability of a map R²→R² weaker than complex differentiability?
- State a sufficient local condition using Cauchy-Riemann equations that avoids the isolated-point trap.
Further reading and review references
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