Liouville's theorem (complex analysis)

In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844[1]), 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 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.

Proof

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This important theorem has several proofs.

A standard analytical proof uses the fact that holomorphic functions are analytic.

Proof

If   is an entire function, it can be represented by its Taylor series about 0:

 

where (by Cauchy's integral formula)

 

and   is the circle about 0 of radius  . Suppose   is bounded: i.e. there exists a constant   such that   for all  . We can estimate directly

 

where in the second inequality we have used the fact that   on the circle  . (This estimate is known as Cauchy's estimate.) But the choice of   in the above is an arbitrary positive number. Therefore, letting   tend to infinity (we let   tend to infinity since   is analytic on the entire plane) gives   for all  . Thus   and this proves the theorem.

Another proof uses the mean value property of harmonic functions.

Proof[2]

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   is bounded, the averages of it over the two balls are arbitrarily close, and so   assumes the same value at any two points.

The proof can be adapted to the case where the harmonic function   is merely bounded above or below. See Harmonic function#Liouville's theorem.

Corollaries

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Fundamental theorem of algebra

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There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem.[3]

No entire function dominates another entire function

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A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if   and   are entire, and   everywhere, then   for some complex number  . Consider that for   the theorem is trivial so we assume  . Consider the function  . It is enough to prove that   can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of   is clear except at points in  . But since   is bounded and all the zeroes of   are isolated, any singularities must be removable. Thus   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 linear

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Suppose that   is entire and  , for  . We can apply Cauchy's integral formula; we have that

 

where   is the value of the remaining integral. This shows that   is bounded and entire, so it must be constant, by Liouville's theorem. Integrating then shows that   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 the complex plane

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

The fact that the domain of a non-constant elliptic function   cannot be   is what Liouville actually proved, in 1847, using the theory of elliptic functions.[4] In fact, it was Cauchy who proved Liouville's theorem.[5][6]

Entire functions have dense images

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If   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   is not dense, then there is a complex number   and a real number   such that the open disk centered at   with radius   has no element of the image of  . Define

 

Then   is a bounded entire function, since for all  ,

 

So,   is constant, and therefore   is constant.

On compact Riemann surfaces

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Any holomorphic function on a compact Riemann surface is necessarily constant.[7]

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.

Remarks

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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   is bounded in a neighborhood of  , then   is a removable singularity of  , i.e.   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   at   —that is, it grows in magnitude comparably to   in some neighborhood of   —then   is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if   for   sufficiently large, then   is a polynomial of degree at most  . This can be proved as follows. Again take the Taylor series representation of  ,

 

The argument used during the proof using Cauchy estimates shows that for all  ,

 

So, if  , then

 

Therefore,  .

Liouville's theorem does not extend to the generalizations of complex numbers known as double numbers and dual numbers.[8]

See also

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References

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  1. ^ Solomentsev, E.D.; Stepanov, S.A.; Kvasnikov, I.A. (2001) [1994], "Liouville theorems", Encyclopedia of Mathematics, EMS Press
  2. ^ Nelson, Edward (1961). "A proof of Liouville's theorem". Proceedings of the American Mathematical Society. 12 (6): 995. doi:10.1090/S0002-9939-1961-0259149-4.
  3. ^ Benjamin Fine; Gerhard Rosenberger (1997). The Fundamental Theorem of Algebra. Springer Science & Business Media. pp. 70–71. ISBN 978-0-387-94657-3.
  4. ^ Liouville, Joseph (1847), "Leçons sur les fonctions doublement périodiques", Journal für die Reine und Angewandte Mathematik, vol. 88 (published 1879), pp. 277–310, ISSN 0075-4102, archived from the original on 2012-07-11
  5. ^ Cauchy, Augustin-Louis (1844), "Mémoires sur les fonctions complémentaires", Œuvres complètes d'Augustin Cauchy, 1, vol. 8, Paris: Gauthiers-Villars (published 1882)
  6. ^ Lützen, Jesper (1990), Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences, vol. 15, Springer-Verlag, ISBN 3-540-97180-7
  7. ^ a concise course in complex analysis and Riemann surfaces, Wilhelm Schlag, corollary 4.8, p.77 http://www.math.uchicago.edu/~schlag/bookweb.pdf Archived 2017-08-30 at the Wayback Machine
  8. ^ Denhartigh, Kyle; Flim, Rachel (15 January 2017). "Liouville theorems in the Dual and Double Planes". Rose-Hulman Undergraduate Mathematics Journal. 12 (2).
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