#4 (ii) Use (i) (otherwise no credit will be given!) to show that cos(20) = 2...
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a)
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% "...
please solve part(ii)
5 Throughout this question the use of a calculator is not permitted. The complex numbers w and z satisfy the relation W= 2 + i iz + 2 (i) Given that z = 1+ i, find w, giving your answer in the form x + iy, where x and y are real. [4] W= (1+i)+i (1+i)+2 1+2ixl-i = Iti Jxl-i 1-1+2i+2 1+12 11 3ti 2 3 2 + li (ii) Given instead that w = z and...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are not shown though the final answer is correct. 1. Show that the real and imaginary parts of the complex-valued function f(x) = cot z are sin 2.c sinh 2 u(x,y) = v(x,y) cos 2. - cosh 2y' cos 2. - cosh 2y (cot z = 1/tanz) [20 points) 2. Obtain the equilibrium points of the following system of 1st or- der ODE and classify...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are not shown though the final answer is correct. 1. Show that the real and imaginary parts of the complex-valued function f(2) = cot z are - sin 2x sinh 2 u(x, y) v(x,y) cos 2.c – cosh 2y' cos 2x - cosh 2y (cot z = 1/ tan ) [20 points)
detailed solution for this one ?????
11. (a) Gi) If w=z+z-' prove that (i) z2 + z 2 = w2 -2 ; 24 +2° + z²+z+1 = z2 (W2 + w+1) = (z? +[1+V5]+1)(22 +[1–V5]+1). (b) Show that the roots of 24 +2+z2+z+1=0 are the four non-real roots of z' =1. (c) Deduce that cos 72° = +(15 – 1) and cos 36° = (15+1).
[20 Marks] Question 2 a) Given f(x)= x - 7x2 +14x-6 i) Show that there is a root a in interval [0,1] (1 mark) ii) Find the minimum number of iterations needed by the bisection method to approximate the root, a of f(x) = 0 on [0,1] with accuracy of 2 decimal points. (3 marks) iii) Find the root (a) of f(x)= x - 7x² +14x6 on [0,1] using the bisection method with accuracy of 2 decimal points. (6 marks)...
(a) For the circuit of Figure 4, assuming a sinusoidal is(t) (0) Prove that the resonant frequeney is given by o- (3 marks) LC (ii) If the total admittance at resonance is 20 ms (seen by the source) with resonant frequency of wo 5000 rad/s and quality factor of Q-10, calculate the values of R L, C, the bandwidth and half-power frequencies in Hertz. (4 marks) VG and hence show (iii) Derive an expression for the driving point impedance Z(jø)...
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) rectangular coordinates (x,y, z); (ii) cylindrical coordinates (r, 0, 2); (iii) spherical coordinates (p, φ,0). a. Inside the sphere 2 +3+224 and above the conezV b. Inside the sphere x2 + y2 + 22-12 and above the paraboloid z 2 2 + y2. c. Inside the sphere 2,2 + y2 + z2-2 and above the surface z-(z2 + y2)1/4 d. Inside the sphere...
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...
8. 4 marks] (a) Suppose that a is the reflection across the line z y +2-0 and β is the reflection ac oss the line y = 2. i. Show that the compositions a(β(z)) and β(a(z)) have a common fixed point in the intersection of two lines. Find this point. ii. Find complex numbers a, b, c, and d for which a(z) = a5+b and β(z) =cz+d. iii. Classify the compositions a(β(z)) and β(a(z)). (b) Let z = 21 and...