![Question 5. Let f(2) = for z e H4 = {z : Im z > 0}, the open upper half-plane of C. 2+i [2]a) Show that f maps H4 into the op](http://img.homeworklib.com/questions/b1d11f50-c7c3-11eb-a53e-2d77775a8cf2.png?x-oss-process=image/resize,w_560)
Homework Answers
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper ha...
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0 slit along the segment from 0 to i, a nonpolygonal region. (Use the principal square root throughout.) Hint: The desired non-polygonal region can be obtained as a "limit" of a sequence of polygonal regions.)
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0...
5. The problem may be a challenging problem. We define and our goal is to show...
5. The problem may be a challenging problem. We define and our goal is to show that f maps the upper half-plane {z : Im(z) >0) to the unit ball (i) Show that if ż-x + iy, then f(x + yi)-u(z, y) + iv(z, y) where ii) Show that the function maps the real axis y -0 to the unit circle. (Hint: Compute (u(x, 0))2 + (v(,0)2) (Bonus Extra 1 point for the homework grade) (iii) Show that f maps...
2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.) 2. Find all one-to-one analytic...
2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.)
2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.)
5. Prove that f(z) = (2+1/2) is a conformal map from the half-disc {z = x...
5. Prove that f(z) = (2+1/2) is a conformal map from the half-disc {z = x +iy : 2< 1, y >0} to the upper half-plane. (Hint: The equation f(z) = w reduces to the quadratic equation z2 + 2wz +1 = 0, which has two distinct roots in C whenever w # £1. This is certainly the case if WE H.
Activity 2
Show that the transformation w = iz + i maps the right half plane Re(z) ≥ 1 onto theupper half plane Im(w) ≥ 2
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed...
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed half plane H = {2€ C: Im(x) > 0} and that there is a finite constant M with f() < M for all z H. 1. Show that da = f(i) x² +1 +00 2. Show that if o is a point in C with Im(a) > 0, then I (a) Im(a)' 22-2Re(a)x+ lajar (3) deduce sin (Bx) where 870
Exercise 2: Möbius Transformations I (a) [10 points] Denote A := {z € C: |z| <...
Exercise 2: Möbius Transformations I (a) [10 points] Denote A := {z € C: |z| < 1}. Prove the following statement. Every Möbius transformation g: A → A who maps A onto A can be written as 9(2) = e® (2- 20 Zoz – 1 with 0 eR and |zo| < 1. Conversely, each such function maps A onto A. (b) [6 points] Find a Möbius transformation f with f(i) = i, f (0) = 0 and f(-i) = 0....
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which...
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
Let f, g E H(C) be such that |f(z)| < \g(z)| for any z e C....
Let f, g E H(C) be such that |f(z)| < \g(z)| for any z e C. Show that there exists a E D(0,1) such that f(z) = ag(z) for any z E C. (Hint: consider f/g and be careful with the zeros of g.)
Notation: In what follows, let D = {z:z<1} and H = {z: Im(2) >0}, 6. Let...
Notation: In what follows, let D = {z:z<1} and H = {z: Im(2) >0}, 6. Let a CD be nonzero. Show that there is a unique automorphism f of D such that f(a) = 0 and f(0) = a. (Hint: Use Theorem 2.2, Chapter 8, Section 2.1 of Stein- Shakarchi (page 220.)
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0 slit along the segment from 0 to i, a nonpolygonal region. (Use the principal square root throughout.) Hint: The desired non-polygonal region can be obtained as a "limit" of a sequence of polygonal regions.)
Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper half-plane Im(w) > 0...
5. The problem may be a challenging problem. We define and our goal is to show that f maps the upper half-plane {z : Im(z) >0) to the unit ball (i) Show that if ż-x + iy, then f(x + yi)-u(z, y) + iv(z, y) where ii) Show that the function maps the real axis y -0 to the unit circle. (Hint: Compute (u(x, 0))2 + (v(,0)2) (Bonus Extra 1 point for the homework grade) (iii) Show that f maps...
2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.)
2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.)
5. Prove that f(z) = (2+1/2) is a conformal map from the half-disc {z = x +iy : 2< 1, y >0} to the upper half-plane. (Hint: The equation f(z) = w reduces to the quadratic equation z2 + 2wz +1 = 0, which has two distinct roots in C whenever w # £1. This is certainly the case if WE H.
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed half plane H = {2€ C: Im(x) > 0} and that there is a finite constant M with f() < M for all z H. 1. Show that da = f(i) x² +1 +00 2. Show that if o is a point in C with Im(a) > 0, then I (a) Im(a)' 22-2Re(a)x+ lajar (3) deduce sin (Bx) where 870
Exercise 2: Möbius Transformations I (a) [10 points] Denote A := {z € C: |z| < 1}. Prove the following statement. Every Möbius transformation g: A → A who maps A onto A can be written as 9(2) = e® (2- 20 Zoz – 1 with 0 eR and |zo| < 1. Conversely, each such function maps A onto A. (b) [6 points] Find a Möbius transformation f with f(i) = i, f (0) = 0 and f(-i) = 0....
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
Let f, g E H(C) be such that |f(z)| < \g(z)| for any z e C. Show that there exists a E D(0,1) such that f(z) = ag(z) for any z E C. (Hint: consider f/g and be careful with the zeros of g.)
Notation: In what follows, let D = {z:z<1} and H = {z: Im(2) >0}, 6. Let a CD be nonzero. Show that there is a unique automorphism f of D such that f(a) = 0 and f(0) = a. (Hint: Use Theorem 2.2, Chapter 8, Section 2.1 of Stein- Shakarchi (page 220.)
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