Equivalent conditions a holomorphic function must fulfill. I‘m currently studying for my complex analysis exam and there seem to be  many, many different equivalent conditions that a holomorphic function must satisfy (but I’ve never seen all of them in one place). So I tried to write everything I‘ve heard up to this point down. I woul really appreciate it if someone could confirm wether these are true or if some of these things are maybe just a one way implication and not an equivalence.
Let $\Omega \subset \mathbb{C}$ be open, connected and $f: \Omega \rightarrow \mathbb{C} $, then these are equivalent:
1.

*

*f is holomorphic on $\Omega$

*f is partially differentiable and satisfies the C.R. equations Edit: this is wrong it should say f is complex differentiable and satisfies the C.R. equations (see comments)

*$\int_\gamma f(z)dz=0$ for every null-homotopic path $\gamma \subset \Omega$ ($\gamma$ can be contracted continuously to a point/there exists a homotopy between $\gamma$ and the constant path)

*for every closed rectangle or triangle $R \subset \Omega$, $\int_{\partial R} f(z)dz=0$

*For every $z_0 \in \Omega$ exists an open subset $V \subset \Omega$ and an holomorphic $F: V \rightarrow \mathbb{C}$ so that $F′=f|_V$

*f can locally be expressed as a Laurent Series around any point in $\Omega$
These are also equivalent:
2.

*

*There exists an $F: \Omega \rightarrow \mathbb{C}$ so that $F′=f$

*$\int_\gamma f(z)dz=0$ for every path $\gamma \subset \Omega$

*f can be expressed as a Taylor series on $\Omega$

*Any of the statements in 1. holds and $\Omega$ is simply connected

 A: Let us see. $f$ is holomorphic in an open simply connected set $U \subseteq \mathbb{C}$ means that $f$ satisfies the Cauchy-Riemann equation (the necessary condition of differenciability) and has continuous partial derivatives (which, together with CR composes the sufficient condition of differenciability).

*

*$f$ is partially differentiable and satisfies CR equations: Yes, this is equivalent, as long as the partial derivatives are continuous in an open set around the point in question, that is, in this context, continuous in $U$.

*$\int_{\gamma}f(z)dz=0$ in every null-homotopic path: I am unfamiliar with the term "null-homotopy", but if it means closed Jordan curves, yes it is equivalent. Green's theorem guarantees this, because it needs the continuity of partial derivatives in an open set contained in the region of integration and because if $f$ satisfies CR Green's theorem yields $0$.

*for every closed rectangle or triangle,$\int_{\partial R}f(z)dz=0$: Again, equivalent to saying $f$ is holomorphic, by Green's theorem, wich relates the line integral on the boundary with the double integral on the set, and, agan, requires continuous partial derivatives on every open set contained in the region of integration

*for every $z_0$ there exists an $F$ holomorphic on an open V such that $F'=f|_V$: if $f$ is the restriction of the derivative of $F$ to an open set, then $f$ has an antiderivative on $V$. This means that the line integral on a closed Jordan curve is $0$ for every path in $V$, which is equivalent to $f$ being holomorphic. The inverse is also true. $f$ holomorphic $\implies$ $f$ has a antiderivative on an open set $V$ around the point where it is holomorphic.

*$f$ can be expressed as a Laurent series around every point of $U$: this is also equivalent to $f$ being holomorphic. It is one of the wonders of Complex Analysis that differentiability, holomorphicity and analiticity are equivalent. Every holomorphic function has a Laurent series that converges to the function on an open set around the point where it is holomorphic. It is a direct consequence of Cauchy's Theorem (Green's theorem in disguise). This one is harder to show that is equivalent in brief words, but one can see that if this is true then the line integral around a Jordan curve is zero, and $f$ being holomorphic is already needed for Cauchy's Theorem to apply.

*There exists an $F$ such that $F'=f$: This one is of course a shortcut equivalent way of expressing the 4th point

*$f$ has null line integral for every path in $U$: this one is not true unless the path is a Jordan curve, so closed, simple, and seccionally regular.

*$f$ can be expressed as a Taylor series:This is again equivalent by virtue of Green's/Cauchy's theorem.

As for your last point I stress that for everything that has to do with Green's/Cauchy's theorem, the domain has to be simply connected so as to any Jordan curve be homotopic to a point.
EDIT:
I believe the domain has to be simply connected for nearly every point. For the first one, if the domain isn't simply connected and your function satisfies CR at every point in the domain, then the partial derivatives can still be continuous if the domain is open. So indeed, it seems that every other point except for the first one is a slightly stronger definition of holomorphic functions, but they are indeed all equivalent under the condition that $U$ is simply connected. The reason I stated it first is because every important theorem of Complex Analysis always assumes it, for it is a necessary condition for Green's Theorem to apply (on top of which Complex Analysis is built). If the domain isn't simply connected you can't deform Jordan curves continuously into a point, which is vital for Green's Theorem to work, as far as I know, and known examples like $\int_{B(0)_{\epsilon}} \dfrac{1}{z}dz \neq 0$ despite $\mathbb{C}\setminus{0}$ being open and $f$ holomorphic in it, back it up. As for the definitions that rely on Laurent and Taylor series: for Taylor, I have never seen the proof, but every statement I have ever seen supposes simple connectedness; for Laurent, it isn't necessary for it to be simply connected, but it does have to be multiply connected, ie, have finitely many "holes", because it makes use of Cauchy's Theorem to work.
