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

Depends on

• 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
• Analytic Hardy spaces on the unit disc Definition
• Analytic primary matrix functions by Jordan blocks Definition
• Biholomorphic maps between complex domains Definition
• Complex analytic functions as locally representable by convergent power series Definition
• Complex projective space and its holomorphic charts Definition
• Complex sector and bounded analytic semigroup Definition
• Continuous logarithms and continuous arguments along a contour Definition
• Function elements and direct analytic continuation Definition
• Holomorphic and antiholomorphic discrete-series models Definition
• Holomorphic functions on an open subset of ℂᵐ Definition
• Holomorphic germs at a point Definition
• Level-one modular forms and cusp forms Definition
• Locally injective holomorphic maps Definition
• Primary matrix sign for spectra off the imaginary axis Definition
• Separately holomorphic functions Definition
• The complex-step derivative estimator and its component-error model Definition
• The complex-time heat kernel on a proper sector Definition
• The modular group and its action on the upper half-plane Definition
• The Nevanlinna class on the disc Definition
• Analytic and nonsmooth data have geometric and algebraic Chebyshev coefficient decay Example
• 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
• Analytic Poisson integrals are exactly the measures with vanishing negative coefficients Lemma
• Compact support gives an entire Fourier-Laplace transform by slices Lemma
• Dixon's glued function is entire and vanishes at infinity Lemma
• Entire rigidity under Gaussian growth and real-axis decay Lemma
• Gaussian decay gives an entire Fourier-Laplace transform and its growth bound Lemma
• Local separable trace-class determinant construction Lemma
• Primitive and Cauchy theorem for Banach-valued holomorphic maps on star-shaped domains Lemma
• Resolvent identity and holomorphy for a closed operator Lemma
• Separately holomorphic functions vanishing on a real box are zero Lemma
• The compact-support Fourier-Laplace transform is entire in each coordinate Lemma
• The complex derivative at a point is unique Lemma
• The disc trace space is the Hardy boundary space and the Szegő family reproduces H² Lemma
• The filled difference quotient is continuous at its exceptional point and holomorphic away from it Lemma

…and 16 more results.

Dependency tree · two levels

10 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)