Calculus of functions ℂ → ℂ. Holomorphic functions are extraordinarily rigid: once-differentiable implies infinitely-differentiable, satisfies Cauchy's theorem, admits power-series and Laurent expansions. Foundational for integral…
complex-analysis
Holomorphic function
f: U ⊆ ℂ → ℂ is holomorphic iff it is complex-differentiable at every point of U. Equivalently (on open sets): analytic, i.e. locally given…
Cauchy-Riemann equations
A function f = u + iv is holomorphic iff u_x = v_y and u_y = −v_x (and u, v are differentiable). Holomorphicity = a pair of PDEs on the…
Contour integral
∫_γ f(z) dz along a piecewise-smooth curve γ in ℂ. Parameterize γ(t): ∫_γ f dz = ∫_a^b f(γ(t)) γ'(t) dt.
Cauchy integral formula
If f is holomorphic on a domain containing a closed disk D, then for any z inside D, f(z) = (1/2πi) ∮_{∂D} f(w)/(w−z) dw. Implies f has…
Laurent series
A function holomorphic on an annulus r < |z − a| < R expands as Σ_{n=-∞}^∞ c_n (z − a)^n. Encodes singularities: the negative-power…
Residue theorem
∮_γ f dz = 2πi Σ Res(f, z_k), summed over poles z_k enclosed by γ. Evaluates huge classes of real and complex integrals that have no…
Analytic continuation
If two holomorphic functions agree on a set with an accumulation point, they agree on every connected component containing it. Uniquely…
Riemann mapping theorem
Any simply-connected proper open subset of ℂ is conformally equivalent to the unit disk. A uniformization result: all such domains look…
Liouville's theorem (complex analysis)
A bounded entire function is constant. Elegant route to the fundamental theorem of algebra.
Schwarz lemma
If f : 𝔻 → 𝔻 is holomorphic with f(0)=0, then |f(z)| ≤ |z| and |f'(0)| ≤ 1; equality anywhere forces f(z) = e^{iθ}z. Rigidifies…
Meromorphic function
A function on an open U ⊂ ℂ that is holomorphic outside a discrete set of poles; equivalently a holomorphic map U → ℂ ∪ {∞} = ℙ¹.
Riemann surface
A 1-dimensional complex manifold (connected). Uniformisation theorem: every simply-connected Riemann surface is conformally equivalent to…
Monodromy
The change in a multivalued function (or flat-bundle section) upon analytic continuation around a loop. The monodromy representation…
Riemann–Roch theorem
For a compact Riemann surface X of genus g and a divisor D: ℓ(D) − ℓ(K − D) = deg D + 1 − g, where K is the canonical divisor. Extends to…
Picard's theorems (little and great)
Little: a non-constant entire function's image misses at most one value. Great: in any neighbourhood of an essential singularity, f takes…
Maximum modulus principle
If f is holomorphic and non-constant on a connected open U, then |f| has no local maximum in U. Consequence: on a bounded domain, |f|…
Möbius transformation
A fractional linear map f(z) = (az + b)/(cz + d) on the Riemann sphere with ad − bc ≠ 0. Möbius transformations form a group isomorphic to…
Uniformisation theorem
Every simply-connected Riemann surface is conformally equivalent to one of exactly three model spaces: the Riemann sphere ℂ̂, the complex…
Residue theorem
∮_γ f dz = 2πi Σ Res(f,a_k) for γ enclosing isolated singularities a_k. Evaluates real integrals via contour closure.
Argument principle
(1/2πi) ∮_γ f'/f dz = (# zeros) − (# poles) inside γ, counted with multiplicity. Rouché's theorem as corollary.
Rouché's theorem
|g| < |f| on closed γ ⟹ f and f+g have same number of zeros inside γ. Standard tool for locating roots; e.g. fundamental theorem of algebra.
Maximum modulus principle
Non-constant holomorphic f on domain cannot attain local max of |f|. Schwarz lemma, Hadamard three-lines refinements.
Open mapping theorem (holomorphic)
Non-constant holomorphic map sends open sets to open sets. Follows from argument principle / constant-coefficient expansion.
Riemann mapping theorem
Every simply-connected proper open Ω ⊂ ℂ is biholomorphic to 𝔻 = {|z|<1}. Foundation of conformal mapping; Koebe's proof.
Uniformization theorem
Every simply-connected Riemann surface is conformally equivalent to ℂ, 𝔻, or ℂP¹. Classifies all Riemann surfaces by covering.
Picard's theorems (little & great)
Little: non-constant entire f omits at most 1 value in ℂ. Great: at essential singularity, f takes every value, with ≤1 exception, ∞-often.
Weierstrass factorization theorem
Entire function f with zeros {a_n} admits product representation f(z) = e^{g(z)} z^m Π E_{p_n}(z/a_n) with primary factors E_p.
Mittag-Leffler theorem
Meromorphic function on Ω with prescribed principal parts exists. Dual to Weierstrass (prescribed zeros).
Conformal mapping
Angle-preserving map: equivalent to holomorphic with non-vanishing derivative. Bilinear (Möbius) transforms on 𝔻, ℍ.
Harmonic function & potential
Δu = 0 in 2D iff u = Re(f) for holomorphic f locally. Mean-value property, maximum principle, Poisson kernel on disk.
Several complex variables & Hartogs
Hartogs extension: holom on 'shell' extends to ball (n ≥ 2) — no isolated singularities. Oka theory; coherent analytic sheaves.
Morera's theorem
Converse of Cauchy's theorem: if f is continuous on an open set Ω and its integral around every triangle contained in Ω is zero, then f is…
Casorati-Weierstrass theorem
Near an essential singularity z_0 of a holomorphic function f, every punctured neighbourhood of z_0 is mapped to a dense subset of ℂ. …
Hurwitz's theorem (holomorphic zeros)
If holomorphic f_n converge locally uniformly to a non-zero limit f on Ω and f has a zero of order k at z_0, then for n large f_n has…
Montel's theorem (normal families)
A family of holomorphic functions on an open set Ω that is uniformly bounded on each compact subset is normal — every sequence has a…
Koebe quarter theorem
Every normalised univalent (injective and holomorphic) map f from the unit disk into ℂ has image containing the ball B(0, 1/4). Extremal:…
Hadamard's three-lines theorem
If f is bounded and holomorphic on the strip 0 < Re z < 1 and continuous up to the boundary, then log M(θ) = log sup_y |f(θ+iy)| is a…
Phragmén-Lindelöf principle
Strengthens the maximum principle on unbounded domains. On an angular sector of opening π/β, a holomorphic function with…
Runge's approximation theorem
A holomorphic function f on an open U ⊃ K (K compact) can be uniformly approximated on K by rational functions whose poles lie in ℂ \ K. …