# proof of Morera’s theorem

We provide a proof of Morera’s theorem^{}
under the hypothesis^{} that
${\int}_{\mathrm{\Gamma}}f(z)\mathit{d}z=0$
for any circuit^{} $\mathrm{\Gamma}$ contained in $G$.
This is apparently more restrictive, but actually equivalent^{},
to supposing
${\int}_{\partial \mathrm{\Delta}}f(z)\mathit{d}z=0$
for any triangle^{} $\mathrm{\Delta}\subseteq G$,
provided that $f$ is continuous^{} in $G$.

The idea is to prove that $f$ has an antiderivative $F$ in $G$.
Then $F$, being holomorphic in $G$,
will have derivatives^{} of any order in $G$;
but ${F}^{(n)}(z)={f}^{(n-1)}(z)$ for all $z\in G$, $n\in \mathbb{N}$, $n\ge 1$.

First, suppose $G$ is connected^{}.
Then $G$, being open, is also pathwise connected.

Fix ${z}_{0}\in G$. For any $z\in G$ define $F(z)$ as

$$F(z)={\int}_{\gamma ({z}_{0},z)}f(w)\mathit{d}w,$$ | (1) |

where $\gamma ({z}_{0},z)$ is a path entirely contained in $G$ with initial point ${z}_{0}$ and final point $z$.

The function $F:G\to \u2102$ is well defined. In fact, let ${\gamma}_{1}$ and ${\gamma}_{2}$ be any two paths entirely contained in $G$ with initial point ${z}_{0}$ and final point $z$; define a circuit $\mathrm{\Gamma}$ by joining ${\gamma}_{1}$ and $-{\gamma}_{2}$, the path obtained from ${\gamma}_{2}$ by “reversing the parameter direction”. Then by linearity and additivity of integral

$${\int}_{\mathrm{\Gamma}}f(w)\mathit{d}w={\int}_{{\gamma}_{1}}f(w)\mathit{d}w+{\int}_{-{\gamma}_{2}}f(w)\mathit{d}w={\int}_{{\gamma}_{1}}f(w)\mathit{d}w-{\int}_{{\gamma}_{2}}f(w)\mathit{d}w;$$ | (2) |

but the left-hand side is 0 by hypothesis, thus the two integrals on the right-hand side are equal.

We must now prove that ${F}^{\prime}=f$ in $G$. Given $z\in G$, there exists $r>0$ such that the ball ${B}_{r}(z)$ of radius $r$ centered in $z$ is contained in $G$. Suppose $$: then we can choose as a path from $z$ to $z+\mathrm{\Delta}z$ the segment $\gamma :[0,1]\to G$ parameterized by $t\mapsto z+t\mathrm{\Delta}z$. Write $f=u+iv$ with $u,v:G\to \mathbb{R}$: by additivity of integral and the mean value theorem,

$\frac{F(z+\mathrm{\Delta}z)-F(z)}{\mathrm{\Delta}z}$ | $=$ | $\frac{1}{\mathrm{\Delta}z}}{\displaystyle {\int}_{\gamma}}f(w)\mathit{d}w$ | ||

$=$ | $\frac{1}{\mathrm{\Delta}z}}{\displaystyle {\int}_{0}^{1}}f(z+t\mathrm{\Delta}z)\mathrm{\Delta}z\mathit{d}t$ | |||

$=$ | $u(z+{\theta}_{u}\mathrm{\Delta}z)+iv(z+{\theta}_{v}\mathrm{\Delta}z)$ |

for some ${\theta}_{u},{\theta}_{v}\in (0,1)$. Since $f$ is continuous, so are $u$ and $v$, and

$$\underset{\mathrm{\Delta}z\to 0}{lim}\frac{F(z+\mathrm{\Delta}z)-F(z)}{\mathrm{\Delta}z}=u(z)+iv(z)=f(z).$$ |

In the general case, we just repeat the procedure
once for each connected component^{} of $G$.

Title | proof of Morera’s theorem |
---|---|

Canonical name | ProofOfMorerasTheorem |

Date of creation | 2013-03-22 18:53:34 |

Last modified on | 2013-03-22 18:53:34 |

Owner | Ziosilvio (18733) |

Last modified by | Ziosilvio (18733) |

Numerical id | 10 |

Author | Ziosilvio (18733) |

Entry type | Proof |

Classification | msc 30D20 |