# Several complex variables analogue of Cauchy's integral theorem

Cauchy's integral theorem says that the integral of any holomorphic function over a closed path in a simply connected domain is always $$0$$.

Is there a version of Cauch's integral theorem for several complex variables?
I mean, if $$f:\Omega\subset\mathbb{C}^n\rightarrow\mathbb{C}$$ is holomorphic and $$\Omega$$ simply connected, then for any closed path $$\gamma\subset\Omega$$, it holds $$\int_\gamma f=0.$$ Maybe it's a trivial question, but I've never found it anywhere, so I'm starting to think it's wrong.

In one variable, you can define the integral $$\int_\gamma f(z)\,dz$$ by $$\int_a^bf(\gamma(t))\cdot \gamma'(t)\,dt$$ You can approximate it by a Riemann sum $$\sum_{k=0}^{n-1} f(z_k)(z_{k+1}-z_k)$$ If you have a primitive function $$F(z)$$, you can compute the integral as $$F(b)-F(a)$$, where $$a$$ and $$b$$ are the end points of the curve $$\gamma$$.
In several variables, all this fail because of dimensional considerations. Then $$\gamma'(t)$$ and $$z_{k+1}-z_k$$ are vectors instead of complex numbers, and there is no notion of a primitive function without specifying which variable you are differentiating with respect to. Hence the expression $$\int_\gamma f$$, integrating a function along a curve, is not even well defined. This is why we integrate 1-forms instead of functions. In contrast to the situation in one variable, there is no universal 1-form to use here.
In $${\bf C}^2$$, use the coordinates $$(z,w)$$ where $$z=x+iy$$ and $$w=u+iv$$. Consider the closed curve $$\gamma$$ given by $$x^2+u^2=1$$. Note that $$\gamma$$ is contained in $${\bf R}\times {\bf R}\subset {\bf C}\times {\bf C}$$, and that we have no complex structure on $${\bf R}\times {\bf R}$$. We can compute $$\int_\gamma f(z,w)\,dz$$ and $$\int_\gamma f(z,w)\,dw$$, but they reduce to ordinary real line integrals, and there is no reason why Cauchy's integral theorem should hold here. For instance, if $$f(z,w)=w$$ and we use the parametrization $$z=\cos(t)$$, $$w=\sin(t)$$, we get $$\int_\gamma f(z,w)\,dz=\int_\gamma u\,dx = -\int_0^{2\pi}\sin^2(t)\,dt=-\pi$$
• There is nothing special about complex numbers here; the issue is the same in ${\bf R}^n$. You can define 1-dimensional integrals in various ways (for instance, with respect to arc length), but you need to make some choice. I've added an example in ${\bf C}^2$ that shows that integration with respect to one of the coordinates will not satisfy Cauchy's integral theorem. Feb 2, 2022 at 14:32
• Thank you very much. This question came from the problem of finding a $z_m$-primitive of a function of several variables. I tried to integrate over a path in the m-slice of $\mathbb{C}^n$, but I would require the slice to be simply connected to well define the integral (i.e. the integral doesn't depend on the choice of the path). If I would allow paths in the whole $\mathbb{C}^n$, then a Cauchy's integral theorem in several variables would be necessary in order to well define the primitive. Feb 2, 2022 at 14:49