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...

Question

Question

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 uni

math Other-Math

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

Solution Given trat All one-to-one aralytc furctons thamap the upper hatf-plare U onto itselp. Nouw, To transfom the probem i 2- When Ago and :0 Then Hence l+군. Theretore, 20 m-2 ItそJ

Add a comment

Answer 2

Similar Homework Help Questions

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. 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.
Question 5. Let f(2) = for z e H4 = {z : Im z > 0},...

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 open unit disc |2| < 1. Hint: compute |f(2)|² for z e H4. [3]b) Show that ƒ maps the boundary of H onto the boundary of the disc |2| <1 minus one point. What point is missed?

(6 (12 pts). Suppose f is a conformal map of the upper half place one-to-one onto...

(6 (12 pts). Suppose f is a conformal map of the upper half place one-to-one onto itself with f(-1) = 0, f(0) = 2, and f(1) = 8. Find f(i).
Exercise 4. (5 points) Find a conformal mapping (a 1-1 analytic map) from the complement of...

Exercise 4. (5 points) Find a conformal mapping (a 1-1 analytic map) from the complement of the non-negative reals, C \[0,00), onto the unit disc [z< 1.
1. Show that w = 2 + the w-plane. NI maps the half circle | 2...

1. Show that w = 2 + the w-plane. NI maps the half circle | 2 = 1,0 <<n in the z-plane to the line segment -2<u<2 in

Problem 2. (18 points) (a) Find a fractional linear transformation that maps the right half-plane to...

Problem 2. (18 points) (a) Find a fractional linear transformation that maps the right half-plane to the unit disk such that the origin is mapped to -1. (b) A fixed point of a transformation T is one where T(2) = 2. Let T be a fractional linear transformation. Assume T is not the identity map. Show T has a most two fixed points. (c) Let S be a circle and 21 a point not on the circle. Show that there...
1. (а) Using an appropriate contour in the upper half plane, find the integral z-1 dz....

1. (а) Using an appropriate contour in the upper half plane, find the integral z-1 dz. (z - i)(z+3i)2 If the contour was closed in the lower half plane, explain how your (b) residue calculation would change.
(Complex Analysis) The linear mapping wFUz+p, where α, β e C maps the point ZFI+1 to the point wi-i, and the poin to the point w2-1i a) Determine α and β. b) Find the region in the w-plane correspond...

(Complex Analysis) The linear mapping wFUz+p, where α, β e C maps the point ZFI+1 to the point wi-i, and the poin to the point w2-1i a) Determine α and β. b) Find the region in the w-plane corresponding to the upper half-plane Im(z) 20 in 9. the z-plane. Sketch the region in the w-plane. c) Find the region in the w-plane corresponding to the disk Iz 2 in the z-plane d) Find the fixed points of the mapping The...

1 point) Show that Φ(u, u) (Au + 2, u-u, 7u + u) parametrizes the plane 2x -y-z = 4, Then (a) Calculate Tu T,, and n(u, v). þ(D), where D = (u, u) : 0 < u < 9,0 < u < 3. (b) Find the area...

1 point) Show that Φ(u, u) (Au + 2, u-u, 7u + u) parametrizes the plane 2x -y-z = 4, Then (a) Calculate Tu T,, and n(u, v). þ(D), where D = (u, u) : 0 < u < 9,0 < u < 3. (b) Find the area of S (c) Express f(x, y, z in terms of u and v and evaluate Is f(x, y,z) ds. (a) Tu n(u,v)- T, (b) Area(S)- (c) JIs f(z, y,2) ds- 1 point)...

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...

Homework Answers

Add Answer to:
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...

Post as a guest

Earn Coins

Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper ha...

5. Prove that f(z) = (2+1/2) is a conformal map from the half-disc {z = x...

Question 5. Let f(2) = for z e H4 = {z : Im z > 0},...

(6 (12 pts). Suppose f is a conformal map of the upper half place one-to-one onto...

Exercise 4. (5 points) Find a conformal mapping (a 1-1 analytic map) from the complement of...

1. Show that w = 2 + the w-plane. NI maps the half circle | 2...

Problem 2. (18 points) (a) Find a fractional linear transformation that maps the right half-plane to...

1. (а) Using an appropriate contour in the upper half plane, find the integral z-1 dz....

(Complex Analysis) The linear mapping wFUz+p, where α, β e C maps the point ZFI+1 to the point wi-i, and the poin to the point w2-1i a) Determine α and β. b) Find the region in the w-plane correspond...

1 point) Show that Φ(u, u) (Au + 2, u-u, 7u + u) parametrizes the plane 2x -y-z = 4, Then (a) Calculate Tu T,, and n(u, v). þ(D), where D = (u, u) : 0 < u < 9,0 < u < 3. (b) Find the area...

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...

Homework Answers

Add Answer to: 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...

Post as a guest

Earn Coins

Add Answer to:
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...