Cauchy,s theorem in Real plane?

In summary, Cauchy's formula is a famous formula that can be applied to both the complex and real plane. When applied to the real plane, it works because of the exact part of the complex form dz/(z-a). In order for Cauchy's theorem to be applied in the real plane, the denominator should be (x-c)^2+(y-d)^2 instead of sqrt((x-c)^2+(y-d)^2). This theorem can also be related to deRham cohomology and states that among all possible 1 forms with curl equal to zero, the only one that is not the gradient of a function is dtheta.
  • #1
eljose79
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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.
 
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  • #2
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).
 
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  • #3


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.
 

1. What is Cauchy's theorem in the Real plane?

Cauchy's theorem, also known as the Cauchy integral theorem, states that if a function is analytic within a closed contour in the complex plane, then the integral of that function over the contour is equal to zero.

2. What is the importance of Cauchy's theorem in Real plane?

Cauchy's theorem is an important result in complex analysis, as it allows for the evaluation of integrals without having to actually perform the integration. It also provides a way to determine if a function is analytic within a given region.

3. Can Cauchy's theorem be extended to functions of more than one variable?

Yes, Cauchy's theorem can be extended to functions of more than one variable, known as Cauchy's integral formula. It states that for a function that is analytic within a closed contour in the complex plane, the value of the function at any point inside the contour is equal to the average value of the function along the contour.

4. What is the relation between Cauchy's theorem and the Cauchy-Riemann equations?

Cauchy's theorem is closely related to the Cauchy-Riemann equations, which are necessary and sufficient conditions for a function to be analytic. These equations express the relationship between the partial derivatives of a complex-valued function and are used to prove the validity of Cauchy's theorem.

5. How is Cauchy's theorem used in real-world applications?

Cauchy's theorem has various applications in engineering, physics, and mathematics. It is used to solve complex integrals, evaluate residues, and analyze the behavior of functions. It also has applications in fluid dynamics, quantum mechanics, and signal processing.

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