# Liouville's theorem (complex analysis)

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In complex analysis, **Liouville's theorem**, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function *f* for which there exists a positive number *M* such that |*f*(*z*)| ≤ *M* for all *z* in **C** is constant.

The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two 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:

where (by Cauchy's integral formula)

and *C*_{r} 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

where in the second inequality we have used the fact that |*z*|=*r* on the circle *C*_{r}. 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 *a*_{k} = 0 for all *k* ≥ 1. Thus *f*(*z*) = *a*_{0} 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.

### 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. To show this, 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, 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

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 **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 **C** is what Liouville actually proved, in 1847, using the theory of elliptic functions.^{[1]} In fact, it was Cauchy who proved Liouville's theorem.^{[2]}^{[3]}

### Entire functions have dense images

If *f* is a non-constant entire function, then its image is dense in **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 *g*(*z*) = 1/(*f*(*z*) − *w*). Then *g* is a bounded entire function, since

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

## Remarks

Let **C** ∪ {∞} be the one point compactification of the complex plane **C**. In place of holomorphic functions defined on regions in **C**, one can consider regions in **C** ∪ {∞}. Viewed this way, the only possible singularity for entire functions, defined on **C** ⊂ **C** ∪ {∞}, 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 at ∞, i.e. blows up like *z ^{n}* 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*.|

*z*| for |

^{n}*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*,

The argument used during the proof using Cauchy estimates shows that

So, if *k* > *n*,

Therefore, *a _{k}* = 0.