Interesting complex variables problem

In summary, the conversation discusses calculating the area of a bounded domain using Green's theorem and expressing it as A(\Omega) = \frac{1}{2}\int_C x dy-y dx. The use of z(t)=x(t)+iy(t) helps reduce the expression.
  • #1
nicksauce
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Homework Statement


Let [tex]\Omega[/tex] be a bounded domain in C whose boundary is a curve z = z(t), a<=t<=b, and let [tex]A(\Omega)[/tex] be the area of [tex]\Omega[/tex]. Prove that

[tex]A(\Omega) = \frac{1}{2}\int^b_a |z(t)|^2 Im(\frac{z'(t)}{z(t)})dt[/tex]


Homework Equations





The Attempt at a Solution


Not even sure where to start on this. Any tips?
 
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  • #2
Yes. [tex]A(\Omega) = \frac{1}{2}\int_C x dy-y dx[/tex]. That's a well known expression for calculating area using Green's theorem. If you express z(t)=x(t)+iy(t), that's what your expression reduces to.
 
  • #3
Ahh right, thank you
 

Related to Interesting complex variables problem

1. What are complex variables?

Complex variables are mathematical quantities that have both real and imaginary components. They are represented in the form a + bi, where a is the real part and bi is the imaginary part. Complex variables are used to solve problems in various fields of science, including physics, engineering, and mathematics.

2. What makes a complex variables problem interesting?

Complex variables problems can be considered interesting when they involve complex mathematical concepts and require creative thinking to solve. These problems often have multiple approaches and can lead to unexpected solutions, making them challenging and engaging for scientists.

3. How are complex variables used in scientific research?

Complex variables are used in many areas of scientific research, including quantum mechanics, electromagnetism, fluid dynamics, and signal processing. They are also used in data analysis and modeling to study complex systems.

4. Can complex variables be visualized?

Yes, complex variables can be visualized using the complex plane. The real part of a complex variable is represented on the x-axis, while the imaginary part is represented on the y-axis. This allows for the visualization of complex functions and their transformations.

5. What are some common applications of complex variables?

Complex variables have many practical applications, such as in electrical engineering for circuit analysis, in physics for studying wave phenomena, and in control theory for designing systems with complex behavior. They are also used in computer graphics and image processing for creating realistic visual effects.

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