Iowa State, MATH 365

Excerpt: ... 65 Example Consider the Joukowski transformation w = J(z) = We will show that 1 2 z+ 1 z . 1. J(z) maps the unit semircle z = ei , 0 interval [-1, 1]. onto the real 2. J(z) maps the intervals [1, ) and (-, -1] onto themselves. 3. J(z) maps the complement of the upper half of the unit disk := {z | |z| > 1, 0 Arg z } onto the upper half plane . Dr. Tim Huber Carver 456 Math 365 Note that J is conformal on since J (z) = 1 (1 - z-2 ) = 0 for 2 z 1. For z = ei , 0 , we have 1 J(z) = J(ei ) = (ei + e-i ) = cos . 2 As varies from 0 to , J(ei ) varies from 1 to -1, showing that the image of the upper semicircle is the interval [-1, 1]. 2. To determine the images of the semi-infinite intervals [1, ) and (-, -1], we let z = x, and note that J(z) = J(x) = 1 2 x+ 1 x . As x varies in [1, ), J(x) varies in the same interval. Thus, J([1, ) = [1, ). Similarly, J(-, -1]) = (-, -1]. 3. Parts 1 and 2 show that the boundary of J() is the real axis. If we can show that J is one-to one, then the previous Lemma [th ...

Ill. Chicago, MATH 104

Excerpt: ... ve length 2i , and whose lower left vertex is the point (j2i , 2i ). For example, Q0,0 is the square Q considered above. (c) The squares {Qi,j | i, j Z} tile the upper half plane . Draw a picture of this tiling (i.e. draw the squares Qi,j for many values i and j). (d) Show that for each i and j, Qi,j and Q0,0 are congruent regions in the upper half plane model of H2 . That is, nd an isometry T of the upper half plane model of H2 such that T (Q0,0 ) = Qi,j . (3) Finite order isometries. Let T be an isometry of a model of neutral geometry. Suppose that T = I but T n = I for some integer n > 1. Prove that T is either a rotation or a reection. (Use the classication of isometries, and rule out every other possibility.) (4) Conjugation. Exercise 20 on p. 374 of Greenberg. (5) Translations of lines. Exercise 12 on p. 373 of Greenberg. (6) Study for the nal exam. 1 ...

UCSD, MATH 11

Excerpt: ... Tessellation for GL(2,F11) acting on the Finite Upper Half Plane over the field with 11*11 elements ...

Iowa State, MATH 365

Excerpt: ... d imaginary axes. Hence, the image of the upper half-circle is either the first quadrant, or the union of the other three quadrants. Since w1 (i/2) = 4 + 3 i, we see that a point in the upper 5 5 half-circle is mapped to the first quadrant. Hence, w1 maps the upper half circle to the first quadrant. The map w2 (z) = z2 sends the first quadrant to the upper half plane since, if z = rei , w2 (z) = r2 e2i . Thus, the argument and magnitude of each complex number is doubled. Dr. Tim Huber Carver 456 Math 365 Where does w(z) = i-z i+z map the upper half plane ? As before, we check the image of the boundary of the domain. We find the image of three points on the real axis w(-1) = -i, w(0) = 1, w(1) = i. Thus, w maps the line through -1, 0 and 1 to a circle or line through the points -i 1 and i. Thus, the image of the x-axis under w is the unit circle. Hence, the image of the upper half plane under w is either the interior or exterior of the unit circle. Since w(i) = 0, a point in the upper half-plane is mapp ...

Rutgers, MATH 403

Excerpt: ... Math 403, section 5 Name Name Computing a definite integral Name Name April 25, 2001 Your goal here is to compute - x2 1 dx using the Residue Theorem. +x+1 1 We may write 1 f(z) dz where IR is the interval from -R dx = lim R I +x+1 - R to R on the real line. f is an analytic function with two isolated singularities. It is given by a formula: x2 f(z) = . in the 2 The isolated singularities of f are at the complex numbers A = upper half plane and B = in the lower half plane. The type of the isolated singularity at A is (circle one) a pole a removable singularity an essential singularity 3 We can factor the denominator of f and rewrite its formula as follows: f(z) = . 4 The function f has an isolated singularity at A in the upper half plane . Since f has a (insert here a useful and precise descriptive phrase about f's singularity at A), the residue of f at A is easy to compute. That residue is . 5 Suppose SR is the counterclockwise oriented semicircle of radius R in the upper half plane centered at 0. ...

North Texas, MATH 5420

Excerpt: ... The Little Picard Theorem William Cherry April 2009 1 The Picard Modular Function as the Universal Covering Map of C \ {0, 1}. = z C : 0 < Re z < 1 and z - 1 1 > 2 2 . Let be the domain By the Riemann mapping theorem, there exists a unique conf ...

Washington, PHYS 2278

Excerpt: ... Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas) This is a good point to take a brief detour and expand on our previous discussions of complex variables and complex functions of complex variables. In particular, we want to consid ...

Excerpt: ... Math 192, Final December 6, 2007. 2:00-4:30 You are NOT allowed calculators, the text, or any other book or notes except for a one recto/verso letter format sheet. SHOW ALL WORK! Write your name and Lecture/Section number on each booklet you use 1) L ...

E. Kentucky, MATH 409

Excerpt: ... tion that transforms the upper half of x = 4 to the upper half of the circle (O; 5). 6. 7. 8. Find a hyperbolic reflection that transforms the upper half of x = 4 to the upper half of x = 17. Find a hyperbolic reflection that transforms the upper half of (O; 5) to the upper half of (O; 3). Find a hyperbolic reflection that transforms the upper half of (O; 5) to the upper half of (9, 0); 2). 9. Prove that given any two points of the upper half plane , there is a hyperbolic reflection that transforms one onto the other. 10. Prove that given any two intersecting geodesics of the upper half plane there is a hyperbolic reflection that transforms one onto the other. 11. Prove that given any two non intersecting geodesics of the upper half plane there is a hyperbolic reflection that transforms one onto the other. 12. 13(C). Prove that in Figure 7.17 (g, m) = XOM. Write a script that will reflect any two points of the upper half-plane in any bowed geodesic. Use the script of Exercise 1.2.17 to substantiat ...

Toledo, PHY 1540

Excerpt: ... Lecture Notes on Mathematical Methods 2008-09 If f = u + iv is analytic, one can show (exercise) by applying the Cauchy-Riemann conditions to the scalar product of the gradients of u and v that the curves of constant u are perpendicular to the curv ...

Georgia Tech, MATH 4320

Excerpt: ... Math 4320 A. D. Andrew Hour Test 3 19 November 2001 Instructions: 1. You may use the text by Churchill and notes on one 8.5" by 11" sheet of paper. 2. Please begin each problem on a new sheet of paper. 3. Be sure to explain your work and justify your results. 4. Problems count equally. 1. How many roots does 6 z4 + z 3 - 2 z 2 + z - 1 have in the disk z < 1? 2. Use the theory of residues to find the inverse Laplace transform of 1 F(s) = . 2 2 ( s + 1) (s + 2 s + 2) 3. Using techniques analogous to those we used to prove the Poisson Integral Formula for the disk, one can deduce the Poisson Integral Formula for the half plane. Specifically, if F is a bounded, piecewise smooth function on the real line, then 1 U(x,y) = ( t - yF)(t)+ y 2 dt - x 2 is harmonic in the upper half plane and lim y 0 + U (x,y) = F (x) at points x where F is continuous. Use this formula to find a function U(x,y), harmonic in the upper half plane , with 0 U(x,0) = 1 x > 1 x < 1 (Please note that this problem was solved by differe ...