(6 (12 pts). Suppose f is a conformal map of the upper half place one-to-one onto...
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,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...
Problem 1.Use Schwarz - Christoffel Formula Theorem to describe the image of the upper half-plane y 2 0 under the conformal mapping w- f(z) that satisfies the given conditions.C ping w = f(z) that satisfies the given conditions. (Do not try to solve for f(z).) a.) f(z) (z +1)1 b) f,(z) = (z + 1)-1/2(z-1)1/2, f(-1) = 0,f(1) = 1 n12(-1)-14, f(-1)-i,f(0) - o,f(1)-1 Problem 1.Use Schwarz - Christoffel Formula Theorem to describe the image of the upper half-plane y...
Suppose a <b and f is a surjective map from the interval [a, b] onto S = {m: m,n e N}. Recall N = {1,2,3,...}. Prove that (a) There exist I, y € [a, b] such that 2 + y and f(x) = f(y). (b) There exists an ro € [a,b] such that lim f(x) does not exist or does not equal f(ro).
2.(10 pts) Suppose that P5 (2) interpolates function f(0) = -at 6 evenly distributed points 0, ,1 on [0,1]. (i) Find an upper bound (as small as possible) for the error f() – P5(); (ii) Give an upper bound (as small as possible) for the maximum of the error max, f(x) – P5(x)]. O<<<1"
Place the following function onto a K-Map, Group the "ones" find the second of 2 minimum SOP expressions f(w,x,y,z) = m(0,6,8, 9, 10, 11, 13, 14, 15)
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2 1 Let f: R R be...
iill like 8 pts.) 7. Find all solutions of in log 2 [12 pts. 8. Compute z dz where T is the upper half unit circle centered at the origin parameterized in the counter- clockwise direction. 8 pts.) 7. Find all solutions of in log 2 [12 pts. 8. Compute z dz where T is the upper half unit circle centered at the origin parameterized in the counter- clockwise direction.
8. [6 pts) Find a formula (possibly a piece-wise one) that defines a continuous function f on the interval |-1,2] such that I f(x) dx = " and " f(a) da = 2.