Striking applications of Morera's theorem

Morera's theorem is an underappreciated theorem in complex analysis. I have been struck by the simplicity of its proof and some clever applications of it and I had been interested in finding out more of such. Please contribute examples. One example is the Weierstrass theorem that if a sequence of holomorphic functions converge absolutely and uniformly on every compact subset in a domain, then the limit is also holomorphic. And there are numerous applications of this latter fact.

So, please come ahead and contribute clever and slick applications of Morera's theorem that will impress people!

Solution 1:

• The Gamma function is defined by $$ \Gamma(s) = \int_0^\infty z^{s-1} e^{-z}\;dz $$ for $\operatorname{Re}(s)>1$ (and is then extended by analytic continuation if we can show it's holomorphic in that domain).

  Is this holomorphic? It can be shown via Morera's theorem. For a closed curve $C$ within $\operatorname{Re}(s)>1$, $$ \int_C \Gamma(s)\;ds = \int_C \int_0^\infty z^{s-1} e^{-z}\;dz \;ds. $$ By Fubini's theorem, this can be shown to be equal to $$ \int_0^\infty \int_C z^{s-1} e^{-z}\;ds \;dz = \int_0^\infty e^{-z} \left(\int_C z^{s-1} \;ds\right) \;dz. $$ The inside integral is $0$ by Cauchy's theorem. Therefore the hypothesis of Morera's theorem is satisfied.

  Therefore $\Gamma$ is holomorphic.

• Riemann's $\zeta$ function is defined by $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} $$ for $\operatorname{Re}(s)>1$ (and is then extended by analytic continuation if we can show it's holomorphic in that domain). So do the same thing as above: $$ \int_C \zeta(s)\;ds = \int\limits_C\ \sum_{n=1}^\infty \frac{1}{n^s} \;ds = \sum_{n=1}^\infty \int_C \frac{1}{n^s} \; ds = \sum_{n=1}^\infty 0 = 0. $$ For the second equality I think either Fubini's theorem or Tonelli's theorem can be used. For the third, Cauchy's theorem.

  Finally, we conclude by Morera's theorem that within $\operatorname{Re}(s)>1$, $\zeta$ is holomoprhic.

• I wouldn't be surprised if this also works for some functions defined as pointwise or uniform limits of sequences of other functions, but I don't know of any such instances.

I seem to recall that several years ago, I put the two examples above into Wikipedia's article titled "Morera's theorem".

Solution 2:

Not sure if either of these applications qualify as "striking," but here's my two cents:

In Rudin's "Real and Complex Analysis" (Third Edition), Morera's Theorem assists in the proofs of the following two interesting theorems. (Here, I'm paraphrasing, not quoting.)

Müntz-Szász Theorem: Let $0 < \lambda_1 < \lambda_2 < \cdots$, and let $X = \{1, x^{\lambda_1}, x^{\lambda_2}, \ldots\} \subset C[0,1].$ Then $X$ is dense in $C[0,1]$ if and only if $\sum \frac{1}{\lambda_n} = \infty$.

In proving the reverse implication $(\Leftarrow)$, Rudin invokes Morera's Theorem to show that the function $$f(z) = \int_0^1 t^z\,d\mu(t)$$ is holomorphic in the right half-plane, where $\mu$ is a complex Borel measure concentrated on $(0,1]$.

Theorem 16.8: Let $\Omega \subset \mathbb{C}$ be a region, $L$ a line or circular arc, and suppose $\Omega - L = \Omega_1 \cup \Omega_2$ is the union of two regions. If $f\colon \Omega \to \mathbb{C}$ is continuous in $\Omega$, and is holomorphic in both $\Omega_1$ and $\Omega_2$, then $f$ is holomorphic on $\Omega$.

Morera's Theorem is used (of course) to show that $f$ is holomorphic.