Liouville's theorem (complex analysis)

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For Liouville's theorem in Hamiltonian mechanics, see Liouville's theorem (Hamiltonian).

In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f(z)| ≤ M for all z in C is constant.

The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant.

1 Proof
2 Corollaries
3 Remarks
4 References
5 External links

[edit] Proof

The theorem follows from the fact that holomorphic functions are analytic. Since f is entire, it can be represented by its Taylor series about 0

$f(z) = \sum_{k=0}^\infty a_k z^k$

where (by Cauchy's integral formula)

$a_k = \frac{f^{(k)}}{k!} = {1 \over 2 \pi i} \oint_{C_r} {f( \zeta )\over \zeta ^{k+1}}\,d\zeta$

and C_r is the circle about 0 of radius r > 0. We can estimate directly

$| a_k | \leq \frac{1}{2 \pi} \oint_{C_r} \frac{ | f ( \zeta ) | }{ | \zeta^{k+1} |} \,d\zeta \leq \frac{1}{2 \pi} \oint_{C_r} \frac{ M }{ r^{k+1} } \,d\zeta \leq \frac{M}{r^k},$

where in the second inequality we have invoked the assumption that |f(z)| ≤ M for all z. But the choice of r in the path integrals used, is arbitrary. Therefore, letting r tend to infinity gives a_k = 0 for all k ≥ 1. Thus f(z) = a₀ and this proves the theorem.

[edit] Corollaries

[edit] Fundamental theorem of algebra

There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem.

[edit] No entire function dominates another entire function

A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if f and g are entire, and |f| ≤ |g| everywhere, then f = α.g for some complex number α. To show this, consider the function

$h = \frac{f}{g}\cdot$

It is enough to prove that h is can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of h is clear except at points in g⁻¹(0). Now if g(a) = 0, then f(a) = 0. Analyticity then implies h can be continuously, therefore holomorphically, extended over a. Thus h can be extended over g⁻¹(0) to an entire function.

[edit] Non-constant elliptic functions can not be defined on C

The theorem can also be used to deduce that the domain of a non-constant elliptic function f can not be C. Suppose it was. Then, if a and b are two periods of f such that ^a⁄_b is not real, consider the parallelogram P whose vertices are 0, a, b and a + b. Then the image of f is equal to f(P). Since f is continuous and P is compact, f(P) is also compact and, therefore, it is bounded. So, f is constant.

The fact that the domain of a non-constant elliptic function f can not be C is what Liouville actually proved, in 1847, using the theory of elliptic functions.^[1] In fact, it was Cauchy who proved Liouville's theorem.^[2]

[edit] Entire functions have dense images

If f is a non-constant entire function, then its image is dense. This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of f was not dense, then there would be a complex number w and a real number r > 0 such that the open disk centered at w with radius r would have no element of the image of f. Define g(z) = ¹⁄_{(f(z) − w)}. Then g would be a bounded entire function, since

$(\forall z\in\mathbb{C}):|g(z)|=\frac1{|f(z)-w|}<\frac1r\cdot$

So, g is constant. This is absurd and therefore the image of f is dense.

[edit] Remarks

Let {∞} ∪ C be the one point compactification of the complex plane C. In place of holomorphic functions defined on regions in C, one can consider regions in {∞} ∪ C. Viewed this way, the only possible singularity for entire functions, defined on C ⊂ {∞} ∪ C, is the point ∞. If an entire function f is bounded in a neighborhood of ∞, then ∞ is a removable singularity of f, i.e. f cannot blow up or behave erratically at ∞. In light of the power series expansion, it is not surprising that Liouville's theorem holds.

Similarly, if an entire function has a pole at ∞, i.e. blows up like zⁿ in some neighborhood of ∞, then f is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if |f(z)| ≤ M.|zⁿ| for |z| sufficiently large, then f is a polynomial of degree at most n. This can be proved as follows. Again take the Taylor series representation of f,