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.
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.
Liouville's theorem: Every holomorphic function
is constant.More succinctly, Liouville's theorem states that every bounded entire function must be constant.
is an entire function, it can be represented by its Taylor series about 0: where (by Cauchy's integral formula) and
Another proof uses the mean value property of harmonic functions.
If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume.
is bounded, the averages of it over the two balls are arbitrarily close, and so
The proof can be adapted to the case where the harmonic function
[3] Suppose for the sake of contradiction that there is a nonconstant polynomial
By the extreme value theorem, a continuous function on a closed and bounded set obtains its extreme values, implying that
, and by Liouville's theorem, is constant, which contradicts our assumption that
A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if
can be extended to an entire function, in which case the result follows by Liouville's theorem.
can be extended to an entire bounded function which by Liouville's theorem implies it is constant.
is bounded and entire, so it must be constant, by Liouville's theorem.
is affine and then, by referring back to the original inequality, we have that the constant term is zero.
The theorem can also be used to deduce that the domain of a non-constant elliptic function
The fact that the domain of a non-constant elliptic function
is what Liouville actually proved, in 1847, using the theory of elliptic functions.
[4] In fact, it was Cauchy who proved Liouville's theorem.
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.
Any holomorphic function on a compact Riemann surface is necessarily constant.
In place of holomorphic functions defined on regions in
Viewed this way, the only possible singularity for entire functions, defined on
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
This extended version of Liouville's theorem can be more precisely stated: if
, The argument used during the proof using Cauchy estimates shows that for all