Let $\phi :\Omega \subset %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion \rightarrow %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $ holomorphic function. Define $x:\Omega \subset \rightarrow \mathbb{C} $ by $x\left( z\right) ={Re}\int_{z_{0}}^{z}\phi \left( \xi \right) d\xi $.
The author states: "It is not necessary that the domain $\Omega$ is simply connected, but only that the periods (that is, that the integral along any closed curve) $\phi_k$ are pure imaginary so that the functions $x_k$ are well defined."
Because for $x_k$ is well defined, just enough? I can not understand. For me if the integral is pure imaginary, then the function $x_k$ will be identically zero.
Could someone help me to understand this better?
It doesn't say that the integral from $z_0$ to $z$ is purely imaginary, but that the integral along a closed curve must be purely imaginary.
This means that if you have two different curves that both go from $z_0$ to $z$, the difference of their integrals will equal the integral around a closed curve (namely, out from $z_0$ to $z$ along one curve and back again along the other) will be purely imaginary, and therefore the two integrals have the same real part.