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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
must be constant. That is, every
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
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.


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 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 :, f'(z), =\frac\left, \oint_\fracd\zeta\\leq \frac \oint_ \frac , d \zeta, \leq \frac \oint_ \frac \left, d\zeta\=\frac 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 In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...
''f'' cannot be \Complex. Suppose it was. Then, if ''a'' and ''b'' are two periods of ''f'' such that is not real, consider the
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
''P'' whose vertices are 0, ''a'', ''b'' and ''a'' + ''b''. Then the image of ''f'' is equal to ''f''(''P''). Since ''f'' is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
and ''P'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
, ''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 In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...
''f'' can not 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) = \frac. Then ''g'' is a bounded entire function, since for all , :, g(z), =\frac < \frac. So, ''g'' is constant, and therefore ''f'' is constant.


On compact Riemann surfaces

Any holomorphic function on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
is necessarily constant. 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 \mathbb such that f(\phi^(z)) is holomorphic on the unit disk and has a maximum at \phi(p_0) \in \mathbb, so it is constant, by the maximum modulus principle.


Remarks

Let \Complex \cup \ 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 \. Viewed this way, the only possible singularity for entire functions, defined on \Complex \subset \Complex \cup \, is the point . If an entire function is bounded in a neighborhood of , then is a
removable singularity In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbour ...
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 , : f(z) = \sum_^\infty a_k z^k. The argument used during the proof using Cauchy estimates shows that for all , :, a_k, \leq Mr^. So, if , then :, a_k, \leq \lim_Mr^ = 0. Therefore, . Liouville's theorem does not extend to the generalizations of complex numbers known as double numbers and dual numbers.


See also

* Mittag-Leffler's theorem


References


External links

* * {{MathWorld , urlname= LiouvillesBoundednessTheorem , title= Liouville’s Boundedness Theorem Theorems in complex analysis Articles containing proofs
holomorphic functions In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...