In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.
The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant.
This important theorem has several proofs.
A standard analytical proof uses the fact that holomorphic functions are analytic.
If f is an entire function, it can be represented by its Taylor series about 0:
where (by Cauchy's integral formula)
and Cr is the circle about 0 of radius r > 0. Suppose f is bounded: i.e. there exists a constant M such that for all z. We can estimate directly
where in the second inequality we have used the fact that on the circle Cr. But the choice of r in the above is an arbitrary positive number. Therefore, letting r tend to infinity (we let r tend to infinity since f is analytic on the entire plane) gives ak = 0 for all k ≥ 1. Thus f(z) = a0 and this proves the theorem.
Another proof uses the mean value property of harmonic functions.
Given two points, choose two balls with the given points as centers and of equal radius. If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume. Since f is bounded, the averages of it over the two balls are arbitrarily close, and so f assumes the same value at any two points.
The proof can be adapted to the case where the harmonic function f is merely bounded above or below. See Harmonic function#Liouville's theorem.
Another proof uses the fact that given a Brownian motion in , such that , we have for all . In words, it says that a harmonic function defines a martingale for the Brownian motion. Then a probabilistic coupling argument finishes the proof.
Fundamental theorem of algebraEdit
No entire function dominates another entire functionEdit
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. Consider that for g = 0 the theorem is trivial so we assume Consider the function h = f/g. It is enough to prove that h 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). But since h is bounded and all the zeroes of g are isolated, any singularities must be removable. Thus h can be extended to an entire bounded function which by Liouville's theorem implies it is constant.
If f is less than or equal to a scalar times its input, then it is linearEdit
Suppose that f is entire and |f(z)| is less than or equal to M|z|, for M a positive real number. We can apply Cauchy's integral formula; we have that
where I is the value of the remaining integral. This shows that f′ is bounded and entire, so it must be constant, by Liouville's theorem. Integrating then shows that f is affine and then, by referring back to the original inequality, we have that the constant term is zero.
Non-constant elliptic functions cannot be defined on ℂEdit
The theorem can also be used to deduce that the domain of a non-constant elliptic function f cannot be 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 is what Liouville actually proved, in 1847, using the theory of elliptic functions. In fact, it was Cauchy who proved Liouville's theorem.
Entire functions have dense imagesEdit
If f is a non-constant entire function, then its image is dense in 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 is not dense, then there is a complex number w and a real number r > 0 such that the open disk centered at w with radius r has no element of the image of f. Define
Then g is a bounded entire function, since for all z,
So, g is constant, and therefore f is constant.
On compact Riemann surfacesEdit
Let be holomorphic on a compact Riemann surface . By compactness, there is a point where attains its maximum. Then we can find a chart from a neighborhood of to the unit disk such that is holomorphic on the unit disk and has a maximum at , so it is constant, by the maximum modulus principle.
Let be the one point compactification of the complex plane In place of holomorphic functions defined on regions in , one can consider regions in Viewed this way, the only possible singularity for entire functions, defined on 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 of order n at ∞—that is, it grows in magnitude comparably to 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,
The argument used during the proof using Cauchy estimates shows that for all k ≥ 0,
So, if k > n, then
Therefore, ak = 0.
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