#### 365lecture20

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 ...

#### hw3

Brookdale, MATH 5020
Excerpt: ... Math 4020/5020 - Analytic Functions Homework #3 Due Friday March 20 1. An arc of 0 radians in a circle is kept at temperature T1 wile the remainder of the circle is kept at T2 . By mapping into the upper half plane , nd the temperature distribution inside the circle. 2. Find a stream function and velocity potential function for ows with the following velocity elds: (a) f = z . (b) f = sin(). z (c) f = z 1 z +1 Note: If f = u + iv the velocity of the ow will be given by the vector (u, v). 1 3. Using the fact that the Joukowski mapping w = z + z maps circles |z| = r > 1 to ellipses, 2 2 nd the complex velocity potential for ow past an ellipse x2 + y2 = 1, where a > b with no a b circulation. 4. Find a Schwartz-Christoel transformation which maps the upper half-plane onto the domain D = {z : 0 < Arg(z) < 4/3} 5. Find the Schwartz-Christoel transformation which maps the upper half-plane onto the triangle with vertices {0, i, 1}. Choose x1 = 1, x2 = 1. This question is ...

#### hw11b

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 ...

#### 27

Johns Hopkins, MATH 443
Excerpt: ... LECTURE 27. THE HEAT EQUATION II FOURIER ANALYSIS (110.443) PROF. QIAO ZHANG Consider the steady-state heat equation in the upper half-plane (*) u(x, y) = 2 u(x, y) 2 u(x, y) + = 0, x2 y 2 1. The Poisson Kernel Let y > 0. Then (1.1) is called the ...

#### 11 tessellation

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

#### Week9

Excerpt: ... incoming energy-momenta E, P and E , p and the same outgoing energy-momenta. Such vertex function is of course symmetric under the particle exchange as described. This concludes the proof of (10.5). Thus we arrive at N = i lim t0 2V dEd3 p log(G) iEt e . (2)4 E (10.10) (10.9) The integral over E seems simple because it is an integral over total derivative. Yet one needs to remember that the presence of the exponential prefactor is important, and the integral needs to be over the contour which closes in the upper half plane . In particular, if G is a free Fermi gas Greens function, G = 1/(E p2 /(2m) + + i0 sign E), then log(G) changes by 2i along this contour if p2 /(2m) < , and by 0 if p2 /(2m) > , leading to n0 (p) to be integrated over p and leading to the standard relation between N/V and pF . In the more general case of the arbitrarily interacting gas, we note that G always has the singularities in the lower half plane if Re E > 0 and in the upper half plane if Re E < ...

#### 365lecture21

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 ...

#### p15408_lect43

Toledo, PHY 1540
Excerpt: ... Lecture Notes on Mathematical Methods 2008-09 4.6 4.6.1 Dispersion relations (BF 6.6) Non-locality in time The response of a system R(t) to a stimulus I(t) cannot be instantaneous. We shall now elucidate important consequences of this fact. The F ...

#### line_int

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. ...

#### 365lecture22

Iowa State, MATH 365
Excerpt: ... Math 365 Complex Analysis Lecture 22 Dr. Tim Huber Carver 456 April 28, 2008 Dr. Tim Huber Carver 456 Math 365 Theorem (Schwarz-Christoffel Transformation) Let be a domain bounded by a polygonal path P with vertices at wj (counted consecutively) and corresponding exterior angles j . Then there is a one to one conformal mapping f (z) of the upper half plane onto such that f (z) = A(z - x1 )-1 / (z - x2 )-2 / (z - xn-1 )-n-1 / , where the xj are real, x1 < x2 < < xn-1 , f (xj ) = wj , lim f (z) = wn , A is constant, and principle branches are used to . define powers; i.e., - < arg z z Dr. Tim Huber Carver 456 Math 365 Dr. Tim Huber Carver 456 Math 365 Sketch of Proof: Refer to Figure 3 on the last page. Suppose that f (z) is a one-to-one conformal mapping such that f (z) = A(z - x1 )-1 /Pi (z - x2 )-2 / (z - xn-1 )-n-1 / . Let wj denote the image of xj under f , i.e., f (xj ) = wj , where x1 < x2 < < xn . Recall that a conformal map f (z) at a point z0 rotates by an angle ...

#### picard

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 ...

#### Chapter10

UNC Charlotte, MATH 6118
Excerpt: ... Chapter 10 Poincar Upper Half Plane Model e The next model of the hyperbolic plane that we will consider is also due to Henri Poincar. e We will be using the upper half plane , or {(x, y) | y > 0}. We will want to think of this with a dierent distance metric on it. Let H = {x + iy | y > 0} together with the arclength element ds = dx2 + dy 2 . y Note that we have changed the arclength element for this model! 10.1 Vertical Lines Let x(t) = (x(t), y(t) be a piecewise smooth parametrization of a curve between the points x(t0 ) and x(t1 ). Recall that in order to nd the length of a curve we break the curve into small pieces and approximate the curve by multiple line segments. In the limiting process we nd that the Euclidean arclength element is ds = dx2 + dy 2 . We then nd the length of a curve by integrating the arclength over the parametrization of the curve. t1 s= t0 dx dt 2 + dy dt 2 dt. Now, we want to work in the Poincar Half Plane model. In this case the length of this e sam ...

#### Phys 228_09_Lec_17

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 ...

#### Phys 228_08_Lec_17

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 ...

#### final questions

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 ...

#### hyp-map

University of Illinois, Urbana Champaign, M 302
Excerpt: ... Math 302 Maps of the Hyperbolic Plane (Revised 3-7-01) The Rectangular Coordinate Map Let H2 be the hyperbolic plane of radius . This means that the radius of the inner circle of each annular piece in your model is . Let E2 be the standard Euclidean ...

#### Euclid7

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 ...

UCSC, MATH 207
Excerpt: ... Math 207 HW7: 1. Page 200: 2,3 2. Page 248: 5, 10, 16 a, 3. Find a fractional linear transformation that maps the points z = 0, 1, i into = 1, i, 0. 4. Let z0 be a point in the upper half plane of the complex plane. Show that the bilinear transformation z z0 = ei0 z z0 Maps the upper half plane into the interior of the unit circle. 5. Find a bilinear transformation mappig upper half plane into unit circle, so that z = i is mapped in to = 0 and the point at is mapped into = 1 1 ...

#### p15408_lect39

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 ...

#### prbset06

Caltech, PH 106
Excerpt: ... Physics 106b Problem set number 6 Due Wednesday, February 24, 1999 Notes about course: Note that there is a \Ph106" box in room 335, which can be used for submitting homework sets. TAs: Yi Li Chiyan Luo Federico Spedalieri lym@its.caltech.edu chiya ...

#### Test3

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 ...

#### ptest2

University of Illinois, Urbana Champaign, M 302
Excerpt: ... or which values of a, b, c and d is f an isometry from E2 to E2 ? Problem 4 (20 points) Let C be a circle of radius 3 centered at (2,0). Let i be inversion through C. (1) Find the image of the point (2,5) under inversion about C. Explain. (2) Find the distortion of i at (2, 5). Explain. (3) Does i take lines to lines? Explain. Problem 5 (20 points) Let U be the upper half plane , and let z : U H2 be the upper half plane map. (1) Explain how to nd z(u, y) for any (u, y) U . (2) Explain how to nd z 1 (P ) for any P H2 . ...