Can Cauchy's Theorem Be Applied in the Real Plane?

[eljose79]

i suppose all of us know the famous Cauchy,s formula

Int(c)f(z)dz/(z-a)=(2pi)if(a)

but could be applied the same to real plane,in fact let be the Integral over the closed path f(x,y)dxdy where f(x,y)=Gradient(g) then the integral in R^2

Int(C)f(x,y)dxdy/[x-a]=?

where [x-a] means sqrt[(x-a)^2+(y-b)^2] could be Cauchy,s theorem be applied in the real plane?..thanks.

 

[mathwonk]

The reason Cauchy's theorem works, stated in real terms is that the complex form dz/(z-a), written out as a real form, has an exact part which gives zero integral around a closed path, plus if(a) times the real form "dtheta" =

-(y-d)/((x-c)^2+(y-d)^2)dx + (x-c)/((x-c)^2 + (y-d)^2) dy,

where a = c + di are the real and imaginary parts of the complex number a.

So I think you do not want to divide by sqrt((x-c)^2+(y-d)^2), rather you may need (x-c)^2+(y-d)^2, as in the denominator of dtheta, to get something interesting.

In fact dtheta, i.e. imaginary part of dz/z, is pretty much the only interesting such integral.

try reading something about deRham cohomology, if you can find anything understandable, maybe Differential topology by Guillemin and Pollack.

here is a theorem: among all possible 1 forms, i.e. expressions Pdx +Qdy, defined in the complement of the origin, among those that have curl equal to zero, i.e. such that dQ/dx = dP/dy, (partial derivatives), essentially the only one that is not the gradient of a function, is dtheta.

I.e. given any such 1 form, there is a function f and a number c such that

Pdx + Qdy = gradf + c(dtheta). Then the integral of both sides over any path going simply once around the origin, equals 2pi c.

I.e. as far as integrating over a circle around the origin, the gradf part gives zero, and hence Pdx +Qdy is equivalent to c(dtheta).

 

[M98Ranger]

Yes, Cauchy's theorem can be applied to the real plane. In fact, Cauchy's integral formula is a generalization of the Fundamental Theorem of Calculus, which is applicable to functions of a single variable. The integral in the real plane can be seen as a special case of the integral in the complex plane, where the imaginary part is equal to zero. Therefore, the same principles and formula can be applied to both cases.

In the real plane, the Cauchy's integral formula would look like this:

∫C f(x,y)dxdy/[x-a]=2πf(a)

where C is a closed path, f(x,y) is a differentiable function, and [x-a] represents the distance between the point (x,y) and the point (a,b). This means that the value of the integral over the closed path is equal to 2π times the value of the function at the point (a,b).

This can be proven using the Cauchy-Riemann equations, which relate the partial derivatives of a function in the real plane to its complex derivative. By applying these equations, we can show that the integral over a closed path in the real plane is equal to the integral over a closed path in the complex plane with the imaginary part set to zero.

In conclusion, Cauchy's theorem can be applied in the real plane, and it is a powerful tool for evaluating integrals over closed paths. It is a fundamental theorem in complex analysis and has many applications in mathematics and physics.