Complex Analysis prerequisite material review

In summary, the set of points satisfying ##1<\vert 2z-6\vert <2## such that ##z\in\Bbb{C}## can be described as ##H\cap O=\{z=x+iy\in\Bbb{C}:\frac{1}{4}<(x-3)^2+y^2<1\}##, where ##H## represents the set of complex numbers at least 1/2 units away from 3 and ##O## represents the set of complex numbers at most 1 unit away from 3. This can also be visualized as an annulus centered at (3,0) with inner radius 1/2 units and outer
  • #1
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Homework Statement


Identify the set of points satisfying ##1<\vert 2z-6\vert <2## such that ##z\in\Bbb{C}##.
My pre-caculus is very rusty, so I am not sure if I am doing this correctly.

Homework Equations


##x^2 +y^2= r^2##
##\forall z,z'\in\Bbb{C}, \vert zz'\vert =\vert z\vert\vert z'\vert##

The Attempt at a Solution


Let ##z=x+iy##. Then ##\vert 2z-6\vert =\vert 2(z-3)\vert=\vert 2\vert\vert z-3\vert## and \begin{align}1<\vert 2z-6\vert<2\Longleftrightarrow \frac{1}{2}<\vert z-3\vert <1\end{align}
We want complex numbers ##z## that are at least ##1/2## units away from ##3## and at most ##1## unit away from ##3##. Let ##H=\{z\in\Bbb{C}:\frac{1}{2}<\vert z-3\vert\}## and ##O=\{z\in\Bbb{C}:\vert z-3\vert <1\}##. We want to find the conditions of all ##z\in\Bbb{C}## such that ##z\in H\cap O##. Since ##\Bbb{R}^2\cong\Bbb{C}##, then the general form of elements in ##H## can be found by solving: ##(x-3)^2 +y^2>(\frac{1}{2})^2\Longrightarrow y>\pm\sqrt{\frac{1}{4}-(x-3)^2}## and for ##O## we solve: ##(x-3)^2+y^2<1^2\Longrightarrow y<\pm\sqrt{1-(x-3)^2}##. Therefore, we have \begin{align}\pm\sqrt{\frac{1}{4}-(x-3)^2}<y< \pm\sqrt{1-(x-3)^2}\Longleftrightarrow 1/4<y<1\end{align}
Hence, ##H\cap O=\{z=x+iy\in\Bbb{C}:\pm\sqrt{\frac{1}{4}-(x-3)^2}<y< \pm\sqrt{1-(x-3)^2}\quad \land\quad 1/4<y<1 \}##
 
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  • #2
You can easily describe the set as an annulus centered at TBD with inner radius TBD and outer radius TBD.
Fill in the blanks and do a sanity check on your answer. You are keeping y away from 0. Does that seem right?

PS. Do you think that the expected answer is in terms of (x,y), or would they expect you just to describe it geometrically in terms of an annulus?
 
  • #3
FactChecker said:
TBD
What is TBD?
 
  • #4
Terrell said:
What is TBD?
On homework problems, I am not allowed to do more than ask leading questions and give hints to redirect you. You should be able to fill in the 'TBD' (To Be Determined) blanks. If not, you may want to think about the problem more in terms of geometric position and distances in the complex plane.
 
  • #5
FactChecker said:
to think about the problem more in terms of geometric position and distances in the complex plane
I think it is all the complex numbers between the circle of radius 1/2 units and circle of radius 1 centered at coordinate (3,0) right?
 
  • #6
Terrell said:
I think it is all the complex numbers between the circle of radius 1/2 units and circle of radius 1 centered at coordinate (3,0) right?
Right. That should make you doubt your answer above, which keeps y away from the x-axis (y=0).

PS. Your original equation 1 is the best algebraic description of the set. I think that may be what they expected as an answer. Or the statement you gave just now in terms of the circles (annulus).
 
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  • #7
FactChecker said:
I think that may be what they expected as an answer.
I am starting to think so too. I'm having difficulty finding a single expression to describe it and I'm not sure if I have forgotten how or there is none.
 
  • #8
can I just simply state that ##H\cap O=\{z=x+iy\in\Bbb{C}:\frac{1}{4}<(x-3)^2+y^2<1\}##? Thanks!
 
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  • #9
Terrell said:
can I just simply state that ##H\cap O=\{z=x+iy\in\Bbb{C}:\frac{1}{4}<(x-3)^2+y^2<1\}##? Thanks!
Good answer.
 
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1. What is Complex Analysis?

Complex Analysis is a branch of mathematics that deals with the study of functions of complex numbers. It combines the concepts of calculus and algebra to analyze the behavior and properties of these functions.

2. What are the prerequisites for studying Complex Analysis?

The main prerequisites for studying Complex Analysis include a strong foundation in calculus, linear algebra, and basic knowledge of complex numbers and their properties. Some knowledge of real analysis and topology may also be helpful.

3. Why is it important to review prerequisite material before studying Complex Analysis?

Reviewing prerequisite material is important because Complex Analysis builds upon concepts from other areas of mathematics. It is essential to have a solid understanding of these concepts in order to fully comprehend and apply the principles of Complex Analysis.

4. What are some recommended resources for reviewing Complex Analysis prerequisites?

Some recommended resources for reviewing Complex Analysis prerequisites include textbooks on calculus, linear algebra, and complex analysis, as well as online courses and video lectures. It is also helpful to practice solving problems and exercises from these resources.

5. How can I prepare for studying Complex Analysis?

To prepare for studying Complex Analysis, it is important to have a strong understanding of the prerequisite material and to practice solving problems. It may also be helpful to familiarize yourself with the basic concepts and notation used in Complex Analysis beforehand.

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