Proving the Area Enclosed by a Simple Closed Path using Complex Analysis

In summary, using Green's theorem, it can be shown that the area enclosed by a positively oriented simple closed piecewise smooth path C is equal to (1/2i)*\int_{C}\bar{z}dz. This can be proven by breaking the contour into real and imaginary parts and using (1/2) integral xdy-ydx. By applying Green's theorem, the integral \int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt can be shown to equal zero, resulting in the desired result.
  • #1
gammy
4
0

Homework Statement


C = positively oriented simple closed piecewise smooth path

Prove that:

(1/2i)*\int_{C}\bar{z}dz

is the area enclosed by C.

Homework Equations



*I know that the curve C is piecewise smooth so that it can be broken up into finitely many pieces so that each piece is smooth.

*Cauchy's integral formula

The Attempt at a Solution



I think that I want to let some function f(t) be a continuous complex-valued function on the path C. Let g(t) be a parametrization of the curve. Frankly, I'm not sure where to go from here. I've just started doing contour integration problems and I have problems knowing where to start with proofs.

I'm just hoping someone can point me in the right direction on where I might want to begin. Thanks.
 
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  • #2
You can compute the area inside a contour using Green's theorem as (1/2) integral xdy-ydx. Break your complex contour into real and imaginary parts.
 
  • #3
A consequence of Green's Theorem is that the area enclosed by C is:

(1/2)\int_C xdy - ydx

which is

(1/2)\int _0 ^1 x(t)y'(t)dt - y(t)x'(t)dt

where g : [0,1] -> C, g(t) = x(t) + iy(t) is your parametrization of C.
 
  • #4
thank you very much for your help.
 
  • #5
Thank you again for the assistance. I have almost gotten to the conclusion that I need but I'm not sure how to proceed.

I have shown that:

(1/2)\int_{0}^{1} \bar{z}dz = (1/2)\int_{0}^{1} x(t)y'(t)dt-y(t)x'(t)dt + (1/2i)\int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt

I am assuming that the integral \int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt is equal to zero because if so I will obtain the result I am looking for, namely that:

(1/2)\int_{0}^{1} \bar{z}dz = (1/2)\int_{0}^{1} x(t)y'(t)dt-y(t)x'(t)dt

I'm not sure how to evaluate \int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt.

Any ideas would be much appreciated.
 
  • #6
Use Green's theorem on it.
 
  • #7
Thank you sir! Solved! Not sure how to update thread title.
 

1. What is the purpose of a complex analysis proof?

A complex analysis proof is used to demonstrate the validity of a mathematical statement or theorem in the field of complex analysis. It involves applying rigorous logic and mathematical techniques to show that a particular statement is true.

2. What is the difference between a complex analysis proof and a regular mathematical proof?

The main difference between a complex analysis proof and a regular mathematical proof is that complex analysis deals with functions and variables that involve complex numbers, which have both real and imaginary components. This means that complex analysis proofs often require more advanced mathematical techniques and concepts.

3. How do you begin a complex analysis proof?

The first step in a complex analysis proof is to clearly state the theorem or statement that is being proven. Then, it is important to define all the variables and terms involved in the statement. From there, the proof can be broken down into smaller steps, using various mathematical techniques and properties to reach the desired conclusion.

4. What are some common techniques used in complex analysis proofs?

Some common techniques used in complex analysis proofs include the use of Cauchy's integral theorem, the Cauchy-Riemann equations, and the Cauchy integral formula. Other techniques may include contour integration, the residue theorem, and the Cauchy convergence criterion.

5. How can I improve my skills in writing complex analysis proofs?

To improve your skills in writing complex analysis proofs, it is important to have a strong understanding of the fundamental concepts and techniques in complex analysis. Practice is also key, so it is helpful to work through a variety of problems and proofs. Additionally, seeking feedback from peers or a mentor can also help improve your skills and understanding of complex analysis proofs.

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