Is there any parallel in Complex Analysis to a surface integral?

["pi"mp]

I've been trying to work through this and see whether you can take an "area" in the complex plane, have x,y vary in some interval, and integrate complex functions over that "area."

The math doesn't seem to work out; plus intuitively, if you're going to sum up a function in a complex variable z, you better be able to say "I will sum up z from some z_a to z_b varied by a parameter t," but if we're looking at an "area" in the complex plane, we cannot say exactly what z varies from, only what it's components x,y vary.

I even tried two parameters in the plane and that didn't seem to yield. Is the reason because the complex plane is actually at 2D depiction of the 1-dimensional vector space C^1?? Therefore, the idea of a double integral over two parameters makes no sense?

Thanks guys

 

[HallsofIvy]

If you are integrating some function of the complex variable, z, "z_a to z_b varied by a parameter t" then you are are integrating over a path in the complex plane, not an area.

 

["pi"mp]

right that was my point, it seems only a path integral can be used, not a surface integral. Just was hoping for an explanation.

 

[tin2019]

Actually there is such a generalization. The surface integral is actually over dz dz*, where z* is treated as an independent variable. The simple change of coordinates accomplishes this z=x+iy and z*=x-iy. However the problem is that this change seems to be more of a formal transition, and the interpretation eludes me. If anybody finds any pedagogical overview of this, I would be happy to read it.

 

[laonious]

I think the fundamental issue is that the notion of area doesn't really make sense here. If you're dealing with a single complex variable, then the space is 1-dimensional over C. So the only way to get an idea of area is to treat it as R^2 and look at z=x+iy, but then you've changed the problem from one about C to one about R^2.

I think a parallel would be integration of 2-forms over C^2.