please solve part(ii) 5 Throughout this question the use of a calculator is not permitted. The...
Al. Practice with complex numbers: Every complex number z can be written in the form z r + iy where r and y are real; we call r the real part of z, written Re z, and likewise y is the imaginary part of z, y - Im z We further define the compler conjugate of z aszT-iy a) Prove the following relations that hold for any complex numbers z, 21 and 22: 2i Re (2122)(Re z) (Re z2) -...
Problem 2. (15 points) a) Find the real part u(x,y) and imaginary part v(x,y) of f(z) = (1+2i)z+ (i – 1)2 +3 b) Verify if the above function is analytic c) Using Laplace's equation verify if the real part u(x,y) is harmonic.
Here is an example of how to do it. 5. Let t be the inversive transformation defined by Determine the image of each of the following generalized circles under : (a) the extended line E U foo], where E is the line with equation y-x (b) the unit circle . 310 5: Inversive Geometry Problem 7 Let be the inversive transformation defined by 2-2i r(z) = 2. 2+2 Use the strategy to determine the image of each of the following...
a) Find the real part u(x,y) and imaginary part v(x,y) of f(2)= (1+2i )z? + (i – 1)2 +3 b) Verify if the above function is analytic c) Using Laplace's equation verify if the real part u(x,y) is harmonic.
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...
show all working please 10 Given z = 2 – j2 is a root of 2z' - 9z2 + 202 - 8 = 0 find the remaining roots of the equation. Find the real and imaginary parts of z when 1 2 1 2 2 + j3 3 - 2 .. Find z = Z4 + z2z3/(z2+z3) when 2, = 2 +j3, z2 = 3 + j4 and 23 = -5+j12. Find the values of the real numbers x and...
Solve i. and ii. Given the ordinary differential equation: cos(x)y' = sin(x)y + 1 Find the general solution of the given differential equation. ii. Solve the ordinary differential equation: ay' + by = a cos(wx) + Bsen(wx) Where: a, b, a,ß and w are nonzero real constants.
8.5 (A refresher on Möbius transformations, for which a variety of techniques is recommended.) Describe the image of (i) {z : Iz-l| > 1 } under z w z/ (z-2), (ii) {x : 름 < Izl < 1 } under z → w = (22+ 1)/ (z-2), (ii) (z Rez0 under z y w, where (w-1)/(w+1) 2(z-1)/(z+1), (iv) {z : 12-1 < 1, Rez < 0 } under z → (z-29/2, (v) D(0; 1) under z ( -)/(z -2). Find...
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...
The angle between two complex vectors x and y is defined as a = arccos Re(x, y) W(x,x)/(y,y) Recall that Re(z) denotes the real part of a complex number z =a+bi, so Re(z) = a. Find the angle a between the vectors x= / -6 -2i 1-4 – 6i) and y= 1-2 – 2i 1 1-6 - 2i a = arccos Be careful to use the correct product everywhere. This is not the dot product.