# Liouville's theorem (complex analysis)

Last updated

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 ${\displaystyle f}$ for which there exists a positive number ${\displaystyle M}$ such that ${\displaystyle |f(z)|\leq M}$ for all ${\displaystyle z}$ in ${\displaystyle \mathbb {C} }$ is constant. Equivalently, non-constant holomorphic functions on ${\displaystyle \mathbb {C} }$ have unbounded images.

## Contents

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

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

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k}}$

where (by Cauchy's integral formula)

${\displaystyle a_{k}={\frac {f^{(k)}(0)}{k!}}={1 \over 2\pi i}\oint _{C_{r}}{\frac {f(\zeta )}{\zeta ^{k+1}}}\,d\zeta }$

and Cr is the circle about 0 of radius r > 0. Suppose f is bounded: i.e. there exists a constant M such that |f(z)| ≤ M for all z. We can estimate directly

${\displaystyle |a_{k}|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {|f(\zeta )|}{|\zeta |^{k+1}}}\,|d\zeta |\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {M}{r^{k+1}}}\,|d\zeta |={\frac {M}{2\pi r^{k+1}}}\oint _{C_{r}}|d\zeta |={\frac {M}{2\pi r^{k+1}}}2\pi r={\frac {M}{r^{k}}},}$

where in the second inequality we have used the fact that |z| = r 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.

## Corollaries

### Fundamental theorem of algebra

There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem. [2]

### 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. Consider that for g = 0 the theorem is trivial so we assume ${\displaystyle g\neq 0.}$ 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)| is less than or equal to M|z|, for M a positive real number. We can apply Cauchy's integral formula; we have that

${\displaystyle |f'(z)|={\frac {1}{2\pi }}\left|\oint _{C_{r}}{\frac {f(\zeta )}{(\zeta -z)^{2}}}d\zeta \right|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {|f(\zeta )|}{\left|(\zeta -z)^{2}\right|}}|d\zeta |\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {M|\zeta |}{\left|(\zeta -z)^{2}\right|}}\left|d\zeta \right|={\frac {MI}{2\pi }}}$

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 ℂ

The theorem can also be used to deduce that the domain of a non-constant elliptic function f cannot be ${\displaystyle \mathbb {C} .}$ 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 ${\displaystyle \mathbb {C} }$ is what Liouville actually proved, in 1847, using the theory of elliptic functions. [3] In fact, it was Cauchy who proved Liouville's theorem. [4] [5]

### Entire functions have dense images

If f is a non-constant entire function, then its image is dense in ${\displaystyle \mathbb {C} .}$ 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

${\displaystyle g(z)={\frac {1}{f(z)-w}}.}$

Then g is a bounded entire function, since for all z,

${\displaystyle |g(z)|={\frac {1}{|f(z)-w|}}<{\frac {1}{r}}.}$

So, g is constant, and therefore f is constant.

## On compact Riemann surfaces

Any holomorphic function on a compact Riemann surface is necessarily constant. [6]

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

## Remarks

Let ${\displaystyle \mathbb {C} \cup \{\infty \}}$ be the one point compactification of the complex plane ${\displaystyle \mathbb {C} .}$ In place of holomorphic functions defined on regions in ${\displaystyle \mathbb {C} }$, one can consider regions in ${\displaystyle \mathbb {C} \cup \{\infty \}.}$ Viewed this way, the only possible singularity for entire functions, defined on ${\displaystyle \mathbb {C} \subset \mathbb {C} \cup \{\infty \},}$ 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,

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k}.}$

The argument used during the proof using Cauchy estimates shows that for all k 0,

${\displaystyle |a_{k}|\leq Mr^{n-k}.}$

So, if k > n, then

${\displaystyle |a_{k}|\leq \lim _{r\to \infty }Mr^{n-k}=0.}$

Therefore, ak = 0.

Liouville's theorem does not extend to the generalizations of complex numbers known as double numbers and dual numbers. [7]

## Related Research Articles

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere in between the two paths, then the two path integrals of the function will be the same.

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions on the complex coordinate space of n-tuples of complex numbers.

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments s with Re(s) > 1 and q with Re(q) > 0 by

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz.

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions to approximate arbitrary non-vanishing holomorphic functions arbitrarily well.

In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which naturally extends the function f from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of T to the bounded operators.

In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions in D for which the p-norm is finite:

In mathematics, in the area of complex analysis, Nachbin's theorem is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.

In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g. More precisely, given an open set in the complex plane and a function the antiderivative of is a function that satisfies .

In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part.

In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace of the simple, unweighted holomorphic Hilbert space of functions square-integrable over the surface of the unit disc of the complex plane, along with a form of the orthogonal projection from to .

In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case.

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on Lp spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.

## References

1. "encyclopedia of mathematics".
2. Benjamin Fine; Gerhard Rosenberger (1997). The Fundamental Theorem of Algebra. Springer Science & Business Media. pp. 70–71. ISBN   978-0-387-94657-3.
3. Liouville, Joseph (1847), "Leçons sur les fonctions doublement périodiques", Journal für die Reine und Angewandte Mathematik (published 1879), 88, pp. 277–310, ISSN   0075-4102, archived from the original on 2012-07-11
4. Cauchy, Augustin-Louis (1844), "Mémoires sur les fonctions complémentaires", Œuvres complètes d'Augustin Cauchy, 1, 8, Paris: Gauthiers-Villars (published 1882)CS1 maint: discouraged parameter (link)
5. Lützen, Jesper (1990), Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences, 15, Springer-Verlag, ISBN   3-540-97180-7
6. 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