1. Sets in the Complex Plane
The concept and evaluation of functions of complex numbers is an extension of the functions of real numbers. Like real-valued functions, discussions of 𝑓(𝑧) necessarily involves the discussions of various sets. Sets define the domain, range, poles, zeros, etc. of complex-valued functions. So, before we can talk about 𝑓(𝑧), we need a few terms of vocabulary concerning sets to help us talk about them:
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2. Sets in the Complex Plane
A neighborhood of radius 𝒓 of a point 𝒛 𝟎 is the collection of all the points inside a circle of radius 𝑟 centered on the point 𝑧 0 . These are the points satisfying 𝑧− 𝑧 0 <𝑟. A deleted neighborhood of 𝑧 0 consists of a neighborhood with the center point 𝑧 0 deleted from the set . An open set is one in which every member of the set has some neighborhood that consists of only other members of the set. For example, 𝑧+5−3𝑖 <3 is an open set. Every point 𝑧 that satisfies the inequality has a neighborhood (possibly very small) that consists only of other points that also satisfy the inequality. The set 𝑧 ≤3 is not an open set because points lying on the boundary do not have a neighborhood (no matter how small) consisting exclusively of members of the set.
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3. Sets in the Complex Plane
The set 𝑧<3 is an open set since it does not include its own boundary.
The set 𝑧≤3 is not open since it does include its own boundary. Any point on the boundary does not have any neighborhood consisting of only members of the set.
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4. Sets in the Complex Plane
A connected set is one in which any two points of the set can be joined by a some path consisting of straight line segments, all of whose points lie within the set. Connected sets can be described as simply or multiply connected. Simply connected means there are no “holes” in the set. Multiply connected means there are gaps, holes or deleted points, but the set is connected nonetheless.
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5. Sets in the Complex Plane
Im 𝑧 <5
Im 𝑧 ≥5
A connected set
A non-connected set
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6. Sets in the Complex Plane
The set 𝑧 <7 (not shown) is a simply connected set whereas the set 2< 𝑧 <7 is a multiply connected set (“doubly connected” in this case) because the circle 𝑧 ≤2 has been removed from the circle 𝑧 <7 . The remaining points do form a connected set.
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7. Sets in the Complex Plane
A domain–type set is a set that is connected and open. A domain must satisfy the criteria for both openness and connectedness. The use of the word “domain” here is distinct from the meaning of “domain” with regard to functions. The word has different meanings depending on the context in which it is used. It is quite possible for the domain of a function to not be a domain-type set.
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8. Sets in the Complex Plane
A boundary point of a set is a point whose every neighborhood contains at least one point of the set and at least one point not of the set. A boundary point may or may not be part of the set that is bounded. N.B.: An open set can not contain any of its boundary points. An interior point of a set has a neighborhood all of whose members belong to the set. An exterior point has a neighborhood all of whose points are not members of the set.
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9. Sets in the Complex Plane
A bounded set is one whose points can be enclosed within a circle of finite radius. That circle may also enclose points that are not members of the set. For example, the set occupying the square 0<Re 𝑧 <1, 0<Im 𝑧 <1 is bounded because we can put a circle around it. The set Re(𝑧)<7 is not bounded.
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10. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
Re 𝑧 <− (a half plane)
The boundary of this set is Re 𝑧 =− 1 2
(a vertical straight line)
This is an OPEN set
(no boundary points are part of the set)
This is a CONNECTED set
(all points of the set can
Connect to all other points of the set along paths within the set)
It is NOT BOUNDED
(it can not be contained within a finite circle)
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11. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
Re 𝑧 =Im 𝑧+𝑖
Open? No.
Connected? Yes.
Bounded? No.
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12. Sets in the Complex Plane
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13. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
−1<Re 𝑧 ≤Im 𝑧+𝑖
Open? No.
Connected? Yes.
Bounded? No.
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14. Sets in the Complex Plane
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15. Sets in the Complex Plane
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16. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
𝑧−2𝑖 ≤2
Open? No.
Connected? Yes
Bounded? Yes.
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17. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
Assignment: find the equation or inequality for the set.
“All the points inside a circle of radius 1 centered on the number 𝑖.”
Ans. The equation for a circle is particularly easy in the complex plane. One way to do it looks like this: 𝑧− 𝑧 0 =𝑟 This is a circle with radius 𝑟 centered on 𝑧 0 . That’s all you need to specify a circle; center and radius.
We are told that the circle has radius 1 and is centered at 𝑎=0 and 𝑏=1 , so in the form above, 𝑟=1 and 𝑧 0 =𝑖. Put that in the form and the boundary of the set is: 𝑧−𝑖 =1.
But our set is the points inside this circle (not including the boundary),
so the set is 𝑧−𝑖 <1
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18. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
Find the equation or inequality for the set.
“All the points outside a circle with radius 3 center at −1−2𝑖.”
Recall a circle can be described by 𝑧− 𝑧 0 =𝑟 . For our circle, 𝑟=3 and 𝑧 0 =−1−2𝑖 , so the boundary is 𝑧+1+2𝑖 =3 .
Since we want the points outside the circle,
the set is 𝑧+1+2𝑖 >3
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19. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
“All the points except the center within a circle centered at
𝑎=2, 𝑏=−1. The radius is 4. Exclude the points on the circle itself.”
Ans.: The equation for the circle is 𝑧−2+𝑖 =4 and the set of all the points within it is 𝑧−2+𝑖 <4. Easy enough.
But what about deleting the center? We could simply add the restriction “𝑧≠2−𝑖” to the definition of the set and that would be correct. But there is a better way…
Instead, we will think of the deleted central point as a “circle” centered on 𝑧 0 with a radius of 0: 𝑧−2+𝑖 =0 . This is the equation of the point 𝑧=2−𝑖.
The set of all the points on the plane except this one can be described as all the point outside this “circle”: 𝑧−2+𝑖 >0. Put these two inequalities together to get: < 𝑧−2+𝑖 <4
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20. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
0< 𝑧−2+𝑖 <4 (repeated).
This is the form for the region between two concentric circles on the complex plane. (We will see this form a lot.) The general form is:
𝑟< 𝑧− 𝑧 0 <𝑅
where 𝑟 is the radius of the inner circle and 𝑅 is the radius of the outer circle.
This type of region is called an “annulus” or “annular region”.
When the inner “circle” is actually the deleted center point, we simply let 𝑟 = 0.
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21. Examples of sets that might be used in complex analysis:
Sets in the Complex Plane
Examples of sets that might be used in complex analysis:
Assignment: find the equation or inequality for the set.
“All the points in the annular region whose center is −1+3𝑖. The inner radius is 1 and the outer radius is The set includes all the points on the outer circle but not the points on the inner circle.”
Ans. From the previous discussion, we know that 𝑟=1, 𝑅=3.5
and 𝑧 0 =−1+3𝑖. We remember to include the boundary points on the outside circle but not the inner one:
1< 𝑧+1−3𝑖 ≤3.5
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22. Sets in the Complex Plane
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23. Sets in the Complex Plane
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