Liouville's theorem (complex analysis) explained

f for which there exists a positive number M such that |f(z)| \leq M for all z\in\Complex is constant. Equivalently, non-constant holomorphic functions on \Complex 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

This important theorem has several proofs.

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

Another proof uses the mean value property of harmonic functions.

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.

Corollaries

Fundamental theorem of algebra

There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem.^[1]

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|\leq |g| everywhere, then f = \alpha g for some complex number \alpha . Consider that for g =0 the theorem is trivial so we assume g ≠ 0 . 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 linear

Suppose that

f is entire and |f(z)|\leq M|z| , for M> 0 . We can apply Cauchy's integral formula; we have that|f'(z)| = (1)/(2 \pi) \left|\oint _{C _{r}} (f(\zeta))/((\zeta - z) ^{2}) d\zeta\right|\leq (1)/(2 \pi) \oint _{C _{r}} (|f(\zeta)|)/(\left|(\zeta - z) ^{2\right|}) |d \zeta|\leq (1)/(2 \pi) \oint _{C _{r}} (M |\zeta|)/(\left|d\zeta\right| = \left| (\zeta - z) ^{2\right|}) (MI)/(2 \pi)

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 the complex plane

f cannot be \Complex . Suppose it was. Then, if a and b are two periods of f such that \tfrac{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.f cannot be \Complex 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 images

If

f is a non-constant entire function, then its image is dense in \Complex . 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 g(z) = (1)/(f(z) - w) .

Then

g is a bounded entire function, since for all z ,|g(z)| = (1)/(|f(z) - w|) < (1)/(r) .

So,

g is constant, and therefore f is constant.

On compact Riemann surfaces

Any holomorphic function on a compact Riemann surface is necessarily constant.^[2]

Let

f(z) be holomorphic on a compact Riemann surface M . By compactness, there is a point p _{0} \in M where |f(p)| attains its maximum. Then we can find a chart from a neighborhood of p _{0} to the unit disk D such that f(\varphi ^{-1} (z)) is holomorphic on the unit disk and has a maximum at \varphi(p _{0)} \in D , so it is constant, by the maximum modulus principle.

Remarks

Let

\Complex \cup \{ inf ty\} be the one-point compactification of the complex plane \Complex . In place of holomorphic functions defined on regions in \Complex , one can consider regions in \Complex \cup \{ inf ty\} . Viewed this way, the only possible singularity for entire functions, defined on \Complex \subset \Complex \cup \{ inf ty\} , is the point inf ty . If an entire function f is bounded in a neighborhood of inf ty , then inf ty is a removable singularity of f , i.e. f cannot blow up or behave erratically at inf ty . 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 inf ty - that is, it grows in magnitude comparably to z ^{n} in some neighborhood of inf ty - then f is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if |f(z)|\leq M|z| ^{n} 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 ,f(z) = \sum ^{inf ty} _{k =0} a _{k} z ^{k.}

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

k\geq 0 ,|a _{k|} \leq Mr ^{n-k} .

So, if

k > n , then|a _{k|} \leq \lim _{r\toinfty} Mr ^{n-k} = 0 .

Therefore,

a _{k} = 0 .

Liouville's theorem does not extend to the generalizations of complex numbers known as double numbers and dual numbers.^[3]

See also

• Mittag-Leffler's theorem

References

1. Book: Benjamin Fine. Gerhard Rosenberger. The Fundamental Theorem of Algebra. 1997. Springer Science & Business Media. 978-0-387-94657-3. 70–71.
2. a concise course in complex analysis and Riemann surfaces, Wilhelm Schlag, corollary 4.8, p.77 http://www.math.uchicago.edu/~schlag/bookweb.pdf
3. Liouville theorems in the Dual and Double Planes. Rose-Hulman Undergraduate Mathematics Journal. 15 January 2017. 12. 2. Denhartigh. Kyle. Flim. Rachel.