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complex-analysis

Complex differentiability

taught

The complex derivative as a genuine limit of a difference quotient in C -- a much stronger condition than real differentiability, not a restatement of it.

Taught in

Requires

  • complex-continuity required
  • partial-derivatives helpful

    The Cauchy-Riemann equations translate complex differentiability into a partial-derivative condition, but the complex derivative itself is a new, stronger idea.

  • the-jacobian helpful

    Complex differentiability corresponds to the real Jacobian being a scaled rotation -- useful intuition, not a substitute for the complex limit definition.

Required by

Local knowledge map

Local knowledge map

Analytic continuation Cauchy integral formula Conformal mapping Evaluating real integrals with residues Harmonic conjugates Introduction to complex dynamics Path independence and complex antiderivatives Residue theorem Schwarz's lemma The ML inequality The Schwarz-Christoffel transformation Cauchy-Goursat theorem Cauchy-Riemann equations Contour integration Identity theorem for analytic functions Maximum modulus principle Maximum principle for analytic functions Möbius transformations Complex differentiability Change of variables in double integrals Exact differential equations Existence and uniqueness (the Picard-Lindelof… From partial derivatives to the total derivative Line integrals of vector fields and work Pullbacks and change of variables Separation of variables: a first look toward PDEs The Jacobian and local linearization The geometric meaning of a partial derivative Topology of the complex plane Continuity of complex functions Partial derivatives The Jacobian A first look at partial rates Choosing a coordinate system Continuity at a point Limits of complex functions

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Solid arrows are required prerequisites; dashed are recommended or helpful. Lower rows are more foundational. Click any box to jump to that concept.