Parameterizing the boundaries as z(t) usually x(t) and y(t).
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all of q1 please, a complex analysis question for complex numbers etc. 1. (a) Define the...
(Complex analysis) Exercise 6 a) Show that the image of the half-plane y > c (c = const) in the z-plane 1 under the inversion mapping w--s the interior of a circle provided that C0 the inversion mapping w hen0? the inversion mapping w = z when c < 0? b) What is the image of the half-plane y > c (c -const) in the z-plane under c) What is the image of the half-plane y > c (cconst) in...
Please answer all questions and explain every answer. Lab Assignment 1: Complex Numbers 1. Find and plot all values of V-64. (hint: convert-64 to polar coordinates) 2. Prove that sin2z cos2z 1. hint: use the formulas for complex sin and cos) 3. Prove that cosh2z - sin2. (hint: use the formulas for complex sin and cos) 4. Use matlab to calculate each of the following quantities: cosh2π, tani, cos (ir) 5. Evaluate the following expressions and compare to the computer...
Question 5 [15 marks] The complex numbers z and w are such that w = 1 + a, z =-b-, where a and b are real and positive. Given that wz 3-4, find the exact values of a and b. [7 marks] The complex numbers z and w are such that lz|-2, arg (z)--2T, lwl = 5, arg(w) = 4T. Find the exact values of i. The real part of z and the imaginary part of z ii. The modulus...
ei0 : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let ø : R -> U 1. (30) Let R be the group of real numbers under addition, and let U be the map given by e2Tir (r) (i) Prove that d is a homomorphism of groups (ii) Find the kernel of ø. (Don't just write down the definition. You need to describe explicit subset of R.) an real number r for...
Let p(z) be the principal branch of 21-i. Let D* = C\(-00,0) be all the complex numbers except for the non-positive real numbers. (a) (4 points) Find a function which is an antiderivative of p(x) on D". (b) (6 points) Let I be a contour such that (i) I is contained in D* and (ii) the initial point of I' is 1 and the terminal point of I is i. Compute (2)dr. Justify your answers.
Complex Analysis: = Define the function 22 f(z) 22 +1 For each annulus region given below, find the Laurent series of f(z) convergent in the region. (a) 0 < 12 – il < 2 (b) 1 < 121.
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,...
1. The quaternions H are a set of "hyper-complex" numbers of the form p = a + bi-cit dk. where a, b, c, d E R. Like the complex numbers, they can be added, conjugated by sending p ? p = a-bi-cj-dk, and the norm of a quaternion is given by lp-va2? 2 247. To multiply two quaternions, we use the algebra i2 = j2 = k2 =-1 ij=-ji = k jk=-kj = i ki=-ik = j a) Use the...
Complex Analysis: 1 + COS Z Define the function 1 f(2)= (z + 1)2(23 +1) (a) Find all the singularities of f(z) and classify each one as either a removable singulatiry, a pole of order m (and find m), or an essential singularity. (b) Let I = 71+72, where yi and 72 are the directed smooth curves parameterized by TT zi(t) = 2i(1 – 2t), 0 < t < 1 z2(t) = 2eit, 277 < t < 2' respectively. Compute...
linear algebra and complex analysis variables please solve this problem quickly 1+i 1. Write in standard form x+yi. 2. Find the modulus and principal argument of z = 2 + 2/3 i and use it to show z' = -218 3. Give geometrical description of the set {z:2z-il 4} 4. Find the principal argument Arg(z) when a) z = -2-21 b) z=(V3 – )6 5. Find three cubic root of i. 6. Show that f(z) = |z|2 is differentiable at...