Exercise 4. (5 points) Find a conformal mapping (a 1-1 analytic map) from the complement of...
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.
4. Find a conformal equivalence from the open unit disc to the set W = {z : 0 < arg(z) < π/4)
(5). This problem involves the mapping w(z)-,(z + z") between the z-plane and the w-plane. The two parts can be solved independently. 2 (a). Identify all of the values of z for which the mapping w(z) fails to be conformal. In each case, explain why the mapping is not conformal at that value of z. (b). Find the image in the w-plane of the unit circle Iz1, Graph it, label the axes, and label the w-plane points that correspond to...
please solve these two questions completely with steps thank you! 2. Find the image of a horizontal line under the mapping w e Problem 5. Evaluate the following integrals, justifying your procedures. 1. e z, where C is the circle with radius, Centre 1,positively oriented. 2. Let CRbe the circle ll R(R> 1), described in the counterclockwise direction. Show that Log Problem 6. The function g(z) = Vre2 (r > 0,-r < θπ) is analytic in its domain of definition,...
Problem 4. (5 points) Suppose f is analytic on and inside a simple closed curve C. Assume f(x) = 0 for z on C. Show f(2)=0 for all z inside C.
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....
Exercise 4 (15 points) (a) Calculate endz where C is a close simple counter-clockwise curve shown in Figure 4. (15 points) Hint: The function f(z) e is analytic. (you do not have to prove the function is analytic) Solution to Exercise 4 1 2 3 4 -1 Figure 4
QUESTION 4 Which of the following will benefit from map generalization? a. Going from a small scale to large scale map O b. Changing map scale from 1:50,000 to 1:250,000 Oc. Changing map scale from 1:500,000 to 1:250,000 Od. Switching off a layer of an interactive digital map e. Zooming in QUESTION 5 Which of the following mapping actions does NOT lead to map generalization when using an online mapping service? a. Zooming out O b. Swapping a point layer...
find the sum of the product from the complement the following expressiorn 1) F (C+A') (C B'D) MUST SHOW ALL OF YOUR WORK IN K-MAP