Question 1. In this question, for brevity we define an open sector in the complex plane to be a s...
Question 1. In this question, for brevity we define an open sector in the complex plane to be a set {z : α < arg(z) < β} and a closed sector to be a set {z : α < arg(z) β), where 0 < β-a < 2π. Consider the following transcendental elementary functions f(z)=e: cos(z): g(z) =e*cos(z) cos(2); 92 2 In which sectors, if any, do each of these functions decay to zero as 00? Explain your answers and distinguish...
Question 1. In this question, for brevity we define an open sector in the complex plane to be a set {z : α < arg(z) < β} and a closed sector to be a set {z : α < arg(z) β), where 0 < β-a < 2π. Consider the following transcendental elementary functions f(z)=e: cos(z): g(z) =e*cos(z) cos(2); 92 2 In which sectors, if any, do each of these functions decay to zero as 00? Explain your answers and distinguish...
Please write neat and explain thank you. This problem concerns embedding the complex plane C with elements zx iy in the Riemann sphere defined in 3-dimensional space R' with coordinates (X,Y,Z) as the set of points satisfying X2 + Y2+22 = 1, which is known as the unit sphere and denoted by S2,or in the context of stereographic projection of the complex plane into the sphere, often referred to as the extended complex plane and denoted by C. We identify...
plz help me solve the question. plz dont copy anyother wrong answer. Ouestion 2. 2/2 -Throughout this question, z E C \ R and we define do (a) Locate and classify all singularities in the complex plane of Determine any associated residues (b) Evaluate Φ(z) by completing the contour in the upper half-plane. (c) Evaluate Ф(z) by completing the contour in the lower half-plane. (d) Verify that your answers to (b) and (c) are the same. (e) If r e...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equivalence relation with as single equivalence class equal to S An equivalence relation and also a total ordering QUESTION 11 A binary operation on a set S, takes any two elements a,b E S and produces another element c e S. Examples of binary operations include...
I need help with d) please help thank you Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the...
the previous hw question and answer 1. Consider the integral from question 2 of the previous homework assignment: too sin ma dx, and assume that both m and a are positive real numbers. By using an indented contour, evaluate this integral fully. You are allowed to resubmit material submitted as part of the previous assignment if you wish.] 2. (30 marks] Evaluate the following integrals: too sin ma x(x2 + a2) dar, m, a real, a +0. rt eike dx,...