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COMPLEX VARIABLES MATH 407

Texas A&M
GPA 3.6

Staff

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This 4 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 407 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 56 views. For similar materials see /class/226049/math-407-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15
12 Concerning subsets of the complex plane In this section we de ne some special types of subsets of the complex plane they will play a basic role in subsequent analysis De nition 121 Suppose that R is a xed positive number7 and that 20 is a xed complex number i The circle of radius R centred at 20 is de ned by C20R 2 EC 2720 R ii The open disc of radius R centred at 20 is de ned by D20R 2 EC 2720 ltR iii The closed disc of radius R centred at 20 is de ned by D20R z E C 2 7 201 R D20R U CzogR De nition 122 Let A be a subset of the complex plane C The complement of A is the set of all complex numbers that are not in A7 to wit7 AC C A z E C z ii Suppose that A is a nonempty subset of C We say that A is open if for every w 6 A7 there is some positive number Ru the subscript denoting the possible dependence of Ru on w such that DwRw Q A iii A subset A of C is said to be closed if AC is open iv A subset A of C is said to be bounded if there is some positive number A such that A Q 50 A7 239e g A for every 2 E A A set is said to be unbounded if it is not bounded v A subset of the complex plane is said to be compact if it is closed and bounded Remark 123 That the entire complex plane C is an open set follows directly from de nition The empty set is also open vacuously Consequently7 both C and the empty set are also closed because they are complements of each other Although we will not prove this fact7 we remark that these are the only two subsets of C that have this property namely being open and closed ii Suppose that A is a nonempty subset of C The reader will con rm that A is open if and only if for every w 6 A7 there is a positive number Tu such that 5w rm Q A iii Suppose that A is as above Then A is not open if and only if there is some wo E A such that Dw0R Q A for every positive number R7 that is7 Dw0R AC 7 Q for every R gt 0 iv Every disc in the complex plane open or closed is a bounded set A closed disc is a compact set v A subset A of C is unbounded if and only if for every positive number T7 there is an element 2T 6 A such that lle gt T Each of the following sets is unbounded a the real axis7 b C7 c 1 ni n a positive integer Example 124 Suppose that R is a xed positive number7 and that 20 E C We show that D20 R is an open set hence the term open disc Let w E D20 R7 so that Ru R7 10720 gt 0 by de nition lf 2 E Dw Rm7 then the triangle inequality gives the estimate zizollziwwizol ziwllwizolltRwlw720lR 1 This shows that Dw Rm Q D20 B As the choice of iv was arbitrary7 we conclude that D20 R is an open set ii The upper half plane HT 26C Szgt0 is an open set Let w a it E HT7 so that t gt O lfz z iy E Dwt7 then 2 7w lt t7 so Proposition 115 implies the inequality ly 7 t lt t Therefore y t7 t 7 y gt 07 ie z E HT7 and it follows that Dwt Q HT This being true for every w E HT7 we conclude that HT is an open set iii The set D20 R is not open for any choice of 20 E C and R gt 0 lndeed7 if wo is any number with we 7 20 R7 then Dw0 r intersects the complement of D20 R for every choice of r gt O The same argument shows that the circle Czo R is not an open set iv The reader is asked to prove that the set U z z iy z gt 07 y gt 0 is an unbounded7 open subset of the complex plane v The reader will verify that the following sets are closed A z E C 2 R and Bzmiy z O De nition 125 Suppose that A Q C is a nonempty set We say that A is disconnected if there exist open sets U and V such that A Q U U V7 both A O U and A V are nonempty sets7 and A O U O V Q The sets U and V are said to provide a disconnection for A ii A subset A of the complex plane is said to be connected if it is not disconnected Example 126 The set A z z iy my 1 is a disconnected subset of the complex plane The reader will check that the sets U z miy z gt 07 y gt 0 and V z miy z lt 07 y lt 0 are open7 and that they provide a disconnection for A De nition 127 Given complex numbers 2 and w7 we denote by 27w the directed line segment which starts at z and ends at w A polygonal curve is a nite union of line segments 20 21 U 21 22 U U 251725 Note that each segment save the rst begins where the preceding one ends ii Suppose that S is a nonempty subset of C We say that S is polygonally connected if every pair of points in S can be joined by a polygonal curve which is contained entirely in S The relationship between connectedness and polygonal connectedness is brought out in the next pair of results7 the rst of which is stated without proof Theorem 128 Suppose that S is a nonempty subset of the complex plane US is polygonally connected then S is connected Even though a connected set need not be polygonally connected in general7 the the two notions coincide for open sets Theorem 129 Ifa nonempty set S is connected and open then it is polygonally connected Proof Let 20 E S De ne U z E S 2 can be joined to 20 by a polygonal curve lying entirely in S V z E S 2 cannot be joined to 20 by a polygonal curve lying entirely in S 2 lfw E U Q S then the openness of S provides a positive number Ru such that Dw Rm Q S Now if z E Dw Rm then the line segment starting at z and ending at w lies within Dw Rm Q S As w belongs to U it can bejoined to 20 by a polygonal curve contained entirely within S Thus 2 can be joined to 20 by a polygonal curve contained entirely within S This shows that DwRw Q U and hence that U is open We assert that V is also an open set To that end let w E V Q S As before there is a positive number rm such that Dw rm Q S because S is open If this disc is not contained in V then it must contain a point say 2 which is not in V As Dw TM is contained in S 2 must belong to S V Therefore 2 is a point in S which can be joined to 20 by a polygonal curve contained in S Once again as the line segment joining w to z is contained in Dwrw Q S it follows that w can be joined to 20 by a polygonal curve contained in S but this contradicts the assumption that w E V Therefore Dw rm Q V hence V is open Now U and V are subsets of S by de nition so S contains their union as well On the other hand if z is any point in S then it can either be joined to 20 by a polygonal curve contained in S in which case 2 E U or it cannot in which case 2 E V So S is contained in the union of U and V and we conclude that S U U V As U O V is empty S O U O V Q as well As 20 E S and S is open there is a positive number 6 such that D20 6 Q S Every point in this disc can be joined to the centre 20 by a line segment in particular a polygonal curve which lies within the disc hence within S Therefore S O U is nonempty Thus we have a pair of open sets U and V such that S U U V S U 7 Q and S O U O V Q So the connectedness of S ensures that S V must be empty otherwise U and V would provide a disconnection for S As V Q S this means that V must be empty that is there is no point in S which cannot be joined to 20 by a polygonal curve contained in S In other words every point in S can be joined to 20 by a polygonal curve contained entirely in S Finally if 21 and 22 are any two points in S then each of them can be joined to 20 by a polygonal curve contained in S hence 21 and 22 can themselves be joined by a polygonal curve contained entirely in S Thus S is polygonally connected I Theorems 128 and 129 can be combined to give the following Corollary 1210 A nonempty open subset of the complex plane is connected if and only ifit is polygonally connected As sets of the type described above play an important role in complex analysis we shall nd it convenient to give them a name De nition 1211 A nonempty open connected hence polygonally connected subset of the complex plane is called a region Remark 1212 The notions of connectedness and openness are not related to one another in particular it is possible for a set to be open and disconnected or to be connected and not open Hence there is no redundancy in the preceding de nition Example 1213 The right half plane HQ zziy zgt0 is a region ii The set A z z iy y 7 0 is an open set which is not connected Hence it is not a region iii The closed disc 50 1 is k connected hence connected However it is not a region because it is not open iv The open disc D20R is a region for any choice of 20 E C and R gt 0 v Let a and b be xed positive numbers The set Szmiy imilta y ltb is a region This is the interior of a rectangle bounded by z ia and y ib

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