# Complex analysis, deceptively tricky problem.

• turnpages
In summary, the set of all z satisfying |z^2-z|<1 can be found by solving the equation |z^2-z|=1. This can be done by first solving the equation z^2-z=w, which results in the multiple valued function z=(w+1/2)^1/2+1/2. This can be further simplified into a sequence of two simple mappings: u=w+1/2 and v=u^1/2. Finally, the mapping from the v plane to the z plane is z=v+1/2. The region in the w-plane depicted by |w|<1 is the interior of the curve formed by the equation |z^2-z|=1.
turnpages
1. Let z be a complex variable. Describe the set of all z satisfying |z^2-z|<1.[\b]

I have a `brute force' solution, but it's really messy. Without a graphing utility, it would be nearly impossible to graph.

I just computed |z^2-z| in terms of x and y, and solved |z^2-z|=1 in this setting. The points inside this curve, then, satisfy the inequality.

It seems like I'm missing a more elegant solution, but I can't see it.

I thought I might be onto something when I did a change of variables and put the expression into the form |u^2-1/4|<1, but that didn't seem to help much.

Solve:
$$z^{2} - z = w$$
w.r.t. $z$. You get the multiple valued function:
$$z = \frac{1 + (1 + 4 w)^{\frac{1}{2}}}{2} = \left(w + \frac{1}{2}\right)^{\frac{1}{2}} + \frac{1}{2}$$
This can be expressed as sequence of two simple mappings:
$$u = w + \frac{1}{2}$$
$$v = u^{\frac{1}{2}}$$
$$z = v + \frac{1}{2}$$

1. What region in the w-plane doese $|w| < 1$ depict?
2. What is the mapping from the w to the u plane?
3. What is the mapping from the u plane to the v plane? Don't forget that $z^{1/2}$ is a double valued function.
4. What is the mapping from the v plane to the z plane?

## What is Complex Analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It involves the analysis of functions of a complex variable and their properties, such as derivatives and integrals.

## Why is Complex Analysis important?

Complex analysis is important because it provides powerful tools for solving problems in many branches of mathematics, physics, and engineering. It is also essential for understanding and solving problems in fields such as fluid dynamics, electromagnetism, and quantum mechanics.

## What are some applications of Complex Analysis?

Complex analysis has many applications in different fields. Some examples include electrical engineering, where it is used to analyze circuits and signals; economics, where it is used to model financial markets; and physics, where it is used to study fluid flow and quantum mechanics.

## What are some common challenges in solving "deceptively tricky" problems in Complex Analysis?

Some common challenges in solving "deceptively tricky" problems in Complex Analysis include identifying the correct approach to solving the problem, understanding the properties of complex functions and their derivatives, and correctly applying the theorems and concepts of complex analysis.

## How can one improve their problem-solving skills in Complex Analysis?

One can improve their problem-solving skills in Complex Analysis by practicing solving a variety of problems, studying the theorems and concepts in depth, and seeking help from experts or classmates when encountering challenging problems.

• Calculus and Beyond Homework Help
Replies
17
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
703
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
754
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
731
• Calculus and Beyond Homework Help
Replies
9
Views
1K