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Difference between revisions of "Liouville's Theorem (complex analysis)"

(statement and proof)
 
m (does not add up, one R gets canceled there, in other proofs and in parametric form of the contour it also ends up with M/R and not M/R^2)
 
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counterclockwise circle of radius <math>R</math> centered at <math>z_0</math>.  Then
 
counterclockwise circle of radius <math>R</math> centered at <math>z_0</math>.  Then
 
<cmath> \lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i}
 
<cmath> \lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i}
−
\int_C -\frac{f(z)}{(z-z_0)^2}dz \biggr\rvert \le \frac{M}{R^2} .</cmath>
+
\int_C \frac{f(z)}{(z-z_0)^2}dz \biggr\rvert \le \frac{M}{R} .</cmath>
 
Since <math>f</math> is holomorphic on the entire complex plane, <math>R</math> can
 
Since <math>f</math> is holomorphic on the entire complex plane, <math>R</math> can
 
be arbitrarily large.  It follows that <math>f'(z) = 0</math>, for every
 
be arbitrarily large.  It follows that <math>f'(z) = 0</math>, for every
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<cmath> f(B) - f(A) = \int_{A}^B f'(z)dz  = 0 , </cmath>
 
<cmath> f(B) - f(A) = \int_{A}^B f'(z)dz  = 0 , </cmath>
 
so <math>f</math> is constant, as desired.  <math>\blacksquare</math>
 
so <math>f</math> is constant, as desired.  <math>\blacksquare</math>
 +
 +
== Extensions ==
 +
 +
It follows from Liouville's theorem if <math>f</math> is a non-constant [[entire]]
 +
function, then the [[image]] of <math>f</math> is [[dense]] in
 +
<math>\mathbb{C}</math>; that is, for every <math>z_0 \in \mathbb{C}</math>, there exists
 +
some <math>z \in \text{Im}\,f</math> that is arbitrarily close to <math>z_0</math>.
 +
 +
=== Proof ===
 +
 +
Suppose on the other hand that there is some <math>z_0</math> not in the
 +
image of <math>f</math>, and that there is a positive real <math>\epsilon</math> such
 +
that <math>\text{Im}\,f</math> has no point within <math>\epsilon</math> of <math>z_0</math>.  Then
 +
the function
 +
<cmath> g(z) = \frac{1}{f(z) - z_0} </cmath>
 +
is holomorphic on the entire complex plane, and it is bounded by
 +
<math>1/\epsilon</math>.  It is therefore constant.  Therefore <math>f</math> is constant.
 +
<math>\blacksquare</math>
 +
 +
[[Picard's Little Theorem]] offers the stronger result that if
 +
<math>f</math> avoids two points in the plane, then it is constant.  It is
 +
possible for an entire function to avoid a single point, as
 +
<math>\exp(z)</math> avoids 0.
 +
  
 
== See also ==
 
== See also ==
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[[Category:Complex analysis]]
 
[[Category:Complex analysis]]
 +
[[Category:Theorems]]

Latest revision as of 19:30, 16 January 2024

In complex analysis, Liouville's Theorem states that a bounded holomorphic function on the entire complex plane must be constant. It is named after Joseph Liouville. Picard's Little Theorem is a stronger result.

Statement

Let $f : \mathbb{C} \to \mathbb{C}$ be a holomorphic function. Suppose there exists some real number $M \ge 0$ such that $\lvert f(z) \rvert \le M$ for all $z \in \mathbb{C}$. Then $f$ is a constant function.

Proof

We use Cauchy's Integral Formula.

Pick some $z_0 \in \mathbb{C}$; let $C_R$ denote the simple counterclockwise circle of radius $R$ centered at $z_0$. Then \[\lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i} \int_C \frac{f(z)}{(z-z_0)^2}dz \biggr\rvert \le \frac{M}{R} .\] Since $f$ is holomorphic on the entire complex plane, $R$ can be arbitrarily large. It follows that $f'(z) = 0$, for every point $z \in \mathbb{C}$. Now for any two complex numbers $A$ and $B$, \[f(B) - f(A) = \int_{A}^B f'(z)dz = 0 ,\] so $f$ is constant, as desired. $\blacksquare$

Extensions

It follows from Liouville's theorem if $f$ is a non-constant entire function, then the image of $f$ is dense in $\mathbb{C}$; that is, for every $z_0 \in \mathbb{C}$, there exists some $z \in \text{Im}\,f$ that is arbitrarily close to $z_0$.

Proof

Suppose on the other hand that there is some $z_0$ not in the image of $f$, and that there is a positive real $\epsilon$ such that $\text{Im}\,f$ has no point within $\epsilon$ of $z_0$. Then the function \[g(z) = \frac{1}{f(z) - z_0}\] is holomorphic on the entire complex plane, and it is bounded by $1/\epsilon$. It is therefore constant. Therefore $f$ is constant. $\blacksquare$

Picard's Little Theorem offers the stronger result that if $f$ avoids two points in the plane, then it is constant. It is possible for an entire function to avoid a single point, as $\exp(z)$ avoids 0.


See also

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