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
• Complex projective space and its holomorphic charts Definition
• Holomorphic and antiholomorphic discrete-series models Definition
• Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
• Meromorphic differentials, orders and residues Definition
• Path integral of a holomorphic differential on a Riemann surface Definition
• The complex-time heat kernel on a proper sector Definition
• Addition and duplication for ℘ Example
• Hyperbolic distances and geodesics in disc and half-plane Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• Rectangular lattices, real mapping, and inverse elliptic integrals Example
• Shannon reconstruction of a sinc function Example
• The Gaussian integral on the line with a certified sinh-sinh truncation tail Example
• The rank-one cotangent and its conic 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
• A Quantitative Logarithm Near the Identity in SU(2) Lemma
• A quasinilpotent trace-class operator has zero trace Lemma
• Degree two of ℘ and its four branch points Lemma
• Entire rigidity under Gaussian growth and real-axis decay Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• Harmonic conjugates and integral logarithmic-pole monodromy on surfaces Lemma
• Maximum principle on a closed strip for bounded holomorphic functions Lemma
• Principal-branch generating function for backward-Euler convolution quadrature Lemma
• Pullback order formula for a branched holomorphic map Lemma
• Regular exhaustion and Dirichlet solutions on relatively compact surface domains Lemma
• Shifted log-Gamma branch is independent of the shift Lemma
• The boundary arc contribution in the valence computation Lemma
• The complex-time heat kernel is L1-differentiable in its parameter Lemma
• The Jacobi product formula for the discriminant Lemma
• The Lipschitz formula for the reciprocal-power sums Lemma
• The unit-disc estimate for Weierstrass elementary factors Lemma
• Transformation laws of the Jacobi theta function Lemma
• Zero free entire function of exponential type is an exponential Lemma

…and 27 more results.

Dependency tree · two levels

14 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)