Complex analysis - area inside a simple closed curve

In summary, complex analysis is a branch of mathematics that deals with functions of complex numbers, including their properties, behavior, derivatives, and integrals. A simple closed curve is a continuous closed loop in the complex plane that can be described as a continuous function mapping a closed interval onto the curve. Cauchy's integral formula can be used to calculate the area inside a simple closed curve, which is significant in determining the analyticity of a function. The Cauchy integral theorem also relates the area inside a simple closed curve to the analyticity of a function.
  • #1
sweetvirgogirl
116
0
Let C be a simple closed curve. Show that the area enclosed by C is given by 1/2i * integral of conjugate of z over the curve C with respect to z.

the hint says: use polar coordinates

i can prove it for a circle, but i am not sure how to extend it to prove it for any given closed curve
 
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  • #2
How would you express the enclosed by an arbitrary closed curve as a double integral? Can you apply Green's theorem to turn this into a single integral over C?
 
  • #3
To prove the given statement, we first need to understand the concept of a simple closed curve and its area. A simple closed curve is a continuous and non-self-intersecting curve that encloses a finite area. This area can be calculated using the Green's theorem, which states that the area enclosed by a simple closed curve C is equal to half the integral of the conjugate of z over C with respect to z.

Now, to prove this using polar coordinates, we need to express the curve C in terms of polar coordinates. Let C be defined by the equation z = r(t)e^(it), where r(t) is the radius of the curve at any given angle t. We can then express the area enclosed by C as:

A = 1/2 * integral of (r(t)e^(it)) * (r'(t)e^(-it)) dt

= 1/2 * integral of r(t)r'(t) dt

= 1/2 * integral of r(t)r'(t) dt

= 1/2 * integral of r(t) * d(r(t)^2)/dt dt

= 1/2 * integral of d(r(t)^2)/dt dt

= 1/2 * (r(t)^2)|_0^(2π)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2 - r(0)^2)

= 1/2 * (r(2π)^2
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with functions of complex numbers. It includes the study of properties and behavior of complex-valued functions, as well as their derivatives and integrals.

2. What is a simple closed curve?

A simple closed curve is a continuous closed loop in the complex plane that does not intersect itself. It can be described as a continuous function that maps a closed interval onto a closed curve in the complex plane.

3. How do you calculate the area inside a simple closed curve?

The area inside a simple closed curve can be calculated using Cauchy's integral formula, which states that the integral of a function along a closed curve is equal to the sum of its residues inside the curve. The residues can be found by evaluating the function at the singular points inside the curve.

4. What is the significance of the area inside a simple closed curve in complex analysis?

The area inside a simple closed curve is an important concept in complex analysis because it is related to the concept of analyticity. A function is said to be analytic if it is differentiable at every point in its domain. The area inside a simple closed curve can be used to determine the analyticity of a function.

5. How is the area inside a simple closed curve related to the Cauchy integral theorem?

The Cauchy integral theorem states that the integral of a function along a closed curve is equal to zero if the function is analytic inside the curve. This means that the area inside a simple closed curve can also be used to determine if a function is analytic or not, and therefore, if the Cauchy integral theorem can be applied.

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