The chain rule for complex derivatives

Depends on

• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\bar z}f=0$, or with the Cauchy–Riemann equations
• The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$

Used by

• A holomorphic logarithm is a primitive of the logarithmic derivative Corollary
• A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order Corollary
• Holomorphic roots of a nonvanishing function on a disc Corollary
• The normalized Riemann map is unique Corollary
• Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
• log|z| has no global harmonic conjugate on C{0 Counterexample
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• FALSE: boundary control alone gives the maximum principle on an unbounded domain False statement
• FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy False statement
• A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• Maximum principle on a closed strip for bounded holomorphic functions Lemma
• The unit-disc estimate for Weierstrass elementary factors Lemma
• Zero free entire function of exponential type is an exponential Lemma
• A bounded harmonic function near an isolated puncture extends harmonically Theorem
• A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
• A plane domain is homologically simply connected exactly when every harmonic function has a global conjugate Theorem
• Equivalent characterisations of a homologically simply connected domain Theorem
• Hadamard three-lines theorem Theorem
• Holomorphic inverse function theorem and local-degree criterion Theorem
• Plane harmonicity is preserved by holomorphic and antiholomorphic changes of coordinate Theorem
• Power maps are biholomorphisms on sectors of width less than 2π/n Theorem
• Schwarz-Pick lemma on the unit disc Theorem

Dependency tree · two levels

12 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, Proposition 2.2.2 (standard reference, not scraped)
• J. Orloff, MIT 18.04 Topic 2, §2.6.1 (standard reference, not scraped)