# Complex Analysis Properties Question 2

• RJLiberator
In summary, the conversation discusses the problem of proving that |e^(i*theta)| = 1 and BAR(e^(i*theta)) = e^(-i*theta). The participants explore the definition of e^(i*theta) and use trigonometric identities to show that both equations are true. They also discuss the parametrization of a point on the unit circle using cos(theta) + isin(theta) and how it relates to the given problem.
RJLiberator
Gold Member
The problem states, Show that:
a) |e^(i*theta)| = 1.
Now, the definition of e^(i*theta) makes this
|cos(theta)+isin(theta)|
If we choose any theta then this should be equal to 1.

What can help me prove this? If I choose, say, pi/6 then it simplifies to |(sqrt(3))/2+i/2)| which doesn't seem to equal 1.

b) BAR(e^(i*theta)) = e^(-i*theta)

Here's what I think. The bar e^(i*theta) means that the definition is cos(theta)-isin(theta) and this can be rewritten as cos(-theta)+i*sin(-theta) which can be inputted back into the definition to see that e^(-i*theta) is correct.

RJLiberator said:
The problem states, Show that:
a) |e^(i*theta)| = 1.
Now, the definition of e^(i*theta) makes this
|cos(theta)+isin(theta)|
If we choose any theta then this should be equal to 1.

What can help me prove this? If I choose, say, pi/6 then it simplifies to |(sqrt(3))/2+i/2)| which doesn't seem to equal 1.

b) BAR(e^(i*theta)) = e^(-i*theta)

Here's what I think. The bar e^(i*theta) means that the definition is cos(theta)-isin(theta) and this can be rewritten as cos(-theta)+i*sin(-theta) which can be inputted back into the definition to see that e^(-i*theta) is correct.

Uh, ##|x+iy|=\sqrt{x^2+y^2}##, yes? So?

RJLiberator
Dick said:
Uh, ##|x+iy|=\sqrt{x^2+y^2}##, yes? So?

Ok, so when we have |cos(theta)+isin(theta)| this can be represented as sqrt(cos^2(theta)+sin^2(theta)) and this is clearly equal to 1, always.

Ah, that is simply beautiful.
I appreciate your guidance here. Swiftly helped me here.

RJLiberator said:
|(sqrt(3))/2+i/2)| which doesn't seem to equal 1.

It does.

Also, I am not sure what the starting point is, what you can assume, but ## cos\theta+ isin\theta ## is a parametrization for a point in the unit circle.

## 1. What is complex analysis and why is it important?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and their properties. Complex numbers are numbers that have both real and imaginary components, and they are used in many areas of mathematics, physics, and engineering. Complex analysis is important because it helps us understand and solve problems involving complex numbers, and it has many applications in fields such as signal processing, fluid dynamics, and quantum mechanics.

## 2. What are the basic properties of complex numbers?

The basic properties of complex numbers include addition, subtraction, multiplication, and division. Complex numbers also have a conjugate property, which means that the conjugate of a complex number is the same number with the sign of the imaginary part changed. Additionally, complex numbers can be represented in polar form, where the magnitude and angle of the number are used to describe it.

## 3. How do we differentiate and integrate complex functions?

To differentiate a complex function, we use the same rules as for real functions, but with the added consideration of the complex components. For example, the derivative of a complex function is found by taking the derivative of the real part and the derivative of the imaginary part separately. To integrate a complex function, we use the Cauchy-Riemann equations, which relate the real and imaginary parts of a complex function to its derivatives.

## 4. What is the Cauchy integral theorem?

The Cauchy integral theorem states that if a function is analytic in a closed region, then the integral of that function around any closed path within that region is equal to zero. This theorem is important in complex analysis because it allows us to evaluate integrals of complex functions by using simpler paths or by breaking the integral into smaller parts.

## 5. How does the concept of analyticity relate to complex analysis?

Analyticity is a key concept in complex analysis, as it refers to functions that are differentiable at every point in their domain. This means that a function is analytic if it has a derivative at every point in its domain. Analytic functions have many important properties, such as being infinitely differentiable and having a unique power series expansion. They also satisfy the Cauchy-Riemann equations, which are essential in the study of complex functions.

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