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We require the expansion for 1W/>4: f(z) = 1 Σ ทะ 4n- (w)nt = 4-2 nz (w)n Now we substitute back Z-21=w: f(z) = 0047-2 , 17-2 (i (2-2))π 12-2;/>4 Now we are going to study one of the important transformations, the simplest after the linear and the inverse transformations, the Möbius transformation. First, let us briefly go over linear transformations. Def 14.1: Function of the form W-Qz+b, a, bET, a&o, is called the linear function (linear transformation) There are 3 important special cases of this transformation. I. Translation: W=Z+b A b z Since the addition of complex numbers. obeys the rules of addition of vectors we get the point w by moving the point z in direction of the vector b to the distance that equals to the length of 6. Obviously, the whole domain I after application of w is moved by vector b. 1 to the Ex 16.1: For W= Z+1: the whole domain is moved by right parallel to x-axis ++w; W = z-zi - the whole domain is moved by I down parallel to y-axis fter; w=2-1+di- 2 Z the whole domain is moved to the direction of the vector -1+ 2i and by distance √ "W 2i di II Rotation: W = eduz, LEIR (!) constant: 1 w/= 121, W= α+ argz. Namely, I is moved to argw= the point w by rotation of the vector z around the origin by angle d. W III Expantion: w= r.z, r>o: 10/= rizl, arg w = argz. The point Z goes to the point w that is on the line connecting Iz with the origin by distance multiplied by r Contraction if O1 Ex 16.2. Points on the circle 1Z1 = 2 under w= 3z pass to (c) show that the image of the annulus {z: ^<12/<2} under 2-1 is the domain { W: Rew >-1, 1-3/31 > 43 2 W = Z-1

We require the expansion for 1W/>4: f(z) = 1 Σ ทะ 4n- (w)nt = 4-2 nz (w)n Now we substitute back Z-21=w: f(z) = 0047-2 , 17-2 (i (2-2))π 12-2;/>4 Now we are going to study one of the important transformations, the simplest after the linear and the inverse transformations, the Möbius transformation. First, let us briefly go over linear transformations. Def 14.1: Function of the form W-Qz+b, a, bET, a&o, is called the linear function (linear transformation) There are 3 important special cases of this transformation. I. Translation: W=Z+b A b z Since the addition of complex numbers. obeys the rules of addition of vectors we get the point w by moving the point z in direction of the vector b to the distance that equals to the length of 6. Obviously, the whole domain I after application of w is moved by vector b. 1 to the Ex 16.1: For W= Z+1: the whole domain is moved by right parallel to x-axis ++w; W = z-zi - the whole domain is moved by I down parallel to y-axis fter; w=2-1+di- 2 Z the whole domain is moved to the direction of the vector -1+ 2i and by distance √ "W 2i di II Rotation: W = eduz, LEIR (!) constant: 1 w/= 121, W= α+ argz. Namely, I is moved to argw= the point w by rotation of the vector z around the origin by angle d. W III Expantion: w= r.z, r>o: 10/= rizl, arg w = argz. The point Z goes to the point w that is on the line connecting Iz with the origin by distance multiplied by r Contraction if O1 Ex 16.2. Points on the circle 1Z1 = 2 under w= 3z pass to (c) show that the image of the annulus {z: ^<12/<2} under 2-1 is the domain { W: Rew >-1, 1-3/31 > 43 2 W = Z-1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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We require the expansion for 1W/>4: f(z) = 1 Σ ทะ 4n- (w)nt = 4-2 nz (w)n Now we substitute back Z-21=w: f(z) = 0047-2 , 17-2 (i (2-2))π 12-2;/>4 Now we are going to study one of the important transformations, the simplest after the linear and the inverse transformations, the Möbius transformation. First, let us briefly go over linear transformations. Def 14.1: Function of the form W-Qz+b, a, bET, a&o, is called the linear function (linear transformation) There are 3 important special cases of this transformation. I. Translation: W=Z+b A b z Since the addition of complex numbers. obeys the rules of addition of vectors we get the point w by moving the point z in direction of the vector b to the distance that equals to the length of 6. Obviously, the whole domain I after application of w is moved by vector b. 1 to the Ex 16.1: For W= Z+1: the whole domain is moved by right parallel to x-axis ++w; W = z-zi - the whole domain is moved by I down parallel to y-axis fter; w=2-1+di- 2 Z the whole domain is moved to the direction of the vector -1+ 2i and by distance √ "W 2i di II Rotation: W = eduz, LEIR (!) constant: 1 w/= 121, W= α+ argz. Namely, I is moved to argw= the point w by rotation of the vector z around the origin by angle d. W III Expantion: w= r.z, r>o: 10/= rizl, arg w = argz. The point Z goes to the point w that is on the line connecting Iz with the origin by distance multiplied by r Contraction if O<r<1, Dilatation if r>1 Ex 16.2. Points on the circle 1Z1 = 2 under w= 3z pass to
Transcribed Image Text:We require the expansion for 1W/>4: f(z) = 1 Σ ทะ 4n- (w)nt = 4-2 nz (w)n Now we substitute back Z-21=w: f(z) = 0047-2 , 17-2 (i (2-2))π 12-2;/>4 Now we are going to study one of the important transformations, the simplest after the linear and the inverse transformations, the Möbius transformation. First, let us briefly go over linear transformations. Def 14.1: Function of the form W-Qz+b, a, bET, a&o, is called the linear function (linear transformation) There are 3 important special cases of this transformation. I. Translation: W=Z+b A b z Since the addition of complex numbers. obeys the rules of addition of vectors we get the point w by moving the point z in direction of the vector b to the distance that equals to the length of 6. Obviously, the whole domain I after application of w is moved by vector b. 1 to the Ex 16.1: For W= Z+1: the whole domain is moved by right parallel to x-axis ++w; W = z-zi - the whole domain is moved by I down parallel to y-axis fter; w=2-1+di- 2 Z the whole domain is moved to the direction of the vector -1+ 2i and by distance √ "W 2i di II Rotation: W = eduz, LEIR (!) constant: 1 w/= 121, W= α+ argz. Namely, I is moved to argw= the point w by rotation of the vector z around the origin by angle d. W III Expantion: w= r.z, r>o: 10/= rizl, arg w = argz. The point Z goes to the point w that is on the line connecting Iz with the origin by distance multiplied by r Contraction if O<r<1, Dilatation if r>1 Ex 16.2. Points on the circle 1Z1 = 2 under w= 3z pass to
(c) show that the image of the annulus {z: ^<12/<2} under 2-1 is the domain { W: Rew >-1, 1-3/31 > 43 2 W = Z-1
Transcribed Image Text:(c) show that the image of the annulus {z: ^<12/<2} under 2-1 is the domain { W: Rew >-1, 1-3/31 > 43 2 W = Z-1
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