Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

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

• A holomorphic logarithm is a primitive of the logarithmic derivative Corollary
• z↦|z|² is complex differentiable exactly at 0, with derivative 0, but is holomorphic on no neighbourhood of 0 Counterexample
• A primitive of a complex function on an open set Definition
• Biholomorphic maps between complex domains Definition
• Complex analytic functions as locally representable by convergent power series Definition
• Continuous logarithms and continuous arguments along a contour Definition
• Function elements and direct analytic continuation Definition
• Holomorphic functions on an open subset of ℂᵐ Definition
• Holomorphic germs at a point Definition
• Locally injective holomorphic maps Definition
• Separately holomorphic functions Definition
• 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
• Dixon's glued function is entire and vanishes at infinity Lemma
• The complex derivative at a point is unique Lemma
• The filled difference quotient is continuous at its exceptional point and holomorphic away from it Lemma
• A holomorphic function of several variables is continuous and separately holomorphic Proposition
• A continuous local inverse has derivative reciprocal to a nonzero complex derivative Theorem
• Cauchy's integral formula for a null-homologous cycle Theorem
• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations Theorem
• Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain Theorem
• Inside its disc of convergence a complex power series is holomorphic and may be differentiated term by term Theorem
• Linearity, product, reciprocal, and quotient rules for complex derivatives Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)