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Liouville's theorem: every bounded entire function is constant
Statement
Every bounded entire function is constant.
More explicitly, if is holomorphic and there is a real such that for every , then is constant.
Facts & Assumptions
Given: An entire function and a real with for every ; the Euclidean identification of as the Euclidean plane and as a normed real algebra: what the identification preserves and the definition of complex domain in A complex domain is a nonempty connected open subset of .
If is holomorphic on , , and on the radius- circle, then for every natural (Cauchy's inequalities bound every derivative by a boundary bound on a compactly contained circle).
A holomorphic function on a complex domain whose derivative vanishes everywhere is constant (A holomorphic function with zero derivative on a domain is constant).
The Euclidean plane is polygonally connected and connected ( is polygonally connected, connected, locally path-connected and locally connected).
Proof
Fix and . Since is holomorphic on and its modulus is at most on the radius- circle, [L1] with derivative order one gives .
Under the identification in the given data, [L3] makes connected; it is also nonempty and open in itself, so it is a complex domain.
If , choose ; then , contradicting step 1.1, so .
Since was arbitrary, step 2.1 gives throughout the domain of step 1.2, and [L2] makes constant; this also covers and every constant entire function.
Depends on
- Cauchy's inequalities bound every derivative by a boundary bound on a compactly contained circle
- A holomorphic function with zero derivative on a domain is constant
- $\mathbb{R}^n$ is polygonally connected, connected, locally path-connected and locally connected
- $\mathbb C=\mathbb R[x]/(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves
- A complex domain is a nonempty connected open subset of $\mathbb C$
Used by
- A bounded holomorphic function on all of ℂᵐ is constant Corollary
- A plane harmonic function bounded above or below is constant Corollary
- Every nonconstant entire function has dense image in the complex plane Corollary
- Compactness and Liouville distinguish the three models Example
- Dixon's gluing traced on the boundary cycle of an annulus Example
- FALSE: an entire function bounded on the real axis is constant False statement
- A quasinilpotent trace-class operator has zero trace Lemma
- The sphere, plane and disc are pairwise biholomorphically distinct Lemma
- Zero free entire function of exponential type is an exponential Lemma
- Addition formula for ℘ Theorem
- Cauchy's integral formula for a null-homologous cycle Theorem
- Divisor and residue laws for elliptic functions Theorem
- Fundamental theorem of algebra by Liouville's theorem Theorem
- Gleason Kahane Zelazko Theorem
- Jacobi theta triple product and nonvanishing of the theta constant Theorem
- Little Picard theorem Theorem
- Meromorphic functions on the Riemann sphere are exactly the rational functions Theorem
- Spectrum is nonempty compact and norm bounded Theorem
- The sphere, the plane, and the disc are pairwise non-biholomorphic Theorem
Dependency tree · two levels
27 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
- Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.3 (standard reference, not scraped)
- E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Corollary 4.5 (standard reference, not scraped)
- Matthias Weber, Complex Analysis, Theorem 2.3.2 (standard reference, not scraped)
- Steven G. Krantz, A Guide to Complex Variables, §3.1.3 (standard reference, not scraped)