Liouville's Theorem (Complex Analysis)/Proof 2
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Theorem
Let $f: \C \to \C$ be a bounded entire function.
Then $f$ is constant.
Proof
By assumption, there is $M \ge 0$ such that $\cmod {\map f z} \le M$ for all $z \in \C$.
Let $r > 0$.
consider:
- $D_r = \set {z \in \C: \cmod z \le r}$
Then, consider the parameterization $\gamma_r : \closedint 0 {2 \pi} \to \partial D_r$ given by:
- $\map {\gamma_r} t := r e^{2 \pi i t}$
For all $z \in \C$, we have:
\(\ds \map {f'} z\) | \(=\) | \(\ds \map {g_z'} 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 \pi} \oint_{\partial D_r} \frac {\map {g_z} w} {w^2} \rd w\) | Cauchy Integral Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 \pi} \int_0^{2 \pi} \frac {\map {g_z} {\map {\gamma_r} t} } { {\map {\gamma_r} t}^2} {\map {\gamma_r '} t} \rd t\) | Definition of Complex Contour Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 \pi} \int_0^{2 \pi} \frac {\map {g_z} {\map {\gamma_r} t} } { {\paren {r e^{2 \pi i t} } }^2} {2 \pi i r e^{2 \pi i t} } \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac i r \int_0^{2 \pi} \frac {\map {g_z} {\map {\gamma_r} t} } { e^{2 \pi i t} } \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac i r \int_0^{2 \pi} \frac {\map f { {\map {\gamma_r} t} + z} } { e^{2 \pi i t} } \rd t\) |
where:
- $\map {g_z} w := \map f {w + z}$
Hence, for all $z \in \C$:
\(\ds \cmod {\map {f'} z}\) | \(=\) | \(\ds \cmod {\frac i r \int_0^{2 \pi} \frac {\map f { {\map {\gamma_r} t} + z} } { e^{2 \pi i t} } \rd t}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \frac 1 r \int_0^{2 \pi} \cmod {\frac {\map f {\map {\gamma_r} t + z} } { e^{2 \pi i t} } } \rd t\) | Modulus of Complex Integral | |||||||||||
\(\ds \) | \(\le\) | \(\ds \frac 1 r \int_0^{2 \pi} M \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 \pi M} r\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds 0\) | as $r \to \infty$ |
Thus it follows that $\map {f'} z = 0$ for all $z \in \C$.
By Zero Derivative implies Constant Complex Function, $f$ is constant.
$\blacksquare$
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Source of Name
This entry was named for Joseph Liouville.