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Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions
Definition
Let be open, let , and let . The function is complex differentiable at if the limit
exists in the metric of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane. Its value is the complex derivative of at , denoted . The increments are nonzero and remain in the domain; because is open, all sufficiently small increments are allowed.
The function is holomorphic on when it is complex differentiable at every point of . A function holomorphic on all of is entire. The word analytic is reserved for the local power-series notion.
Depends on
- $\mathbb C=\mathbb R[x]/(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves
- The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
Used by
- z↦|z|² is complex differentiable exactly at 0, with derivative 0, but is holomorphic on no neighbourhood of 0 Counterexample
- z↦ z² is entire with derivative 2z, directly from the complex difference quotient Example
- z↦1/z is holomorphic on ℂ∖{0} with derivative -1/z², directly from the difference quotient Example
- FALSE: a holomorphic function with zero derivative on an arbitrary open set is constant False statement
- FALSE: the Cauchy–Riemann equations at one point imply complex differentiability there False statement
- The complex derivative at a point is unique Lemma
- A continuous local inverse has derivative reciprocal to a nonzero complex derivative Theorem
- Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_bar zf=0, or with the Cauchy–Riemann equations Theorem
- Linearity, product, reciprocal, and quotient rules for complex derivatives Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 59 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Guide to Cultivating Complex Analysis, §2.1.1 (standard reference, not scraped)
- J. Orloff, MIT 18.04 Topic 2, §2.6 (standard reference, not scraped)