Complex Analysis Answers

express cos^(3)tetasin^(4)teta in terms of multiple angle

We have a sample of 50 micro-drills for drilling holes in low-carbon alloy steel. The average lifetime, expressed as the number of holes drilled before failure, is 12.68. The standard deviation of the sample of 50 is 6.83 holes. Find the 95% confidence interval for this sample.

QUESTION 9


9.1 State De Moivre’s Theorem.

(2)


9.2 Express cos 5θ and sin 4θ as polynomials in terms of sin θ and cos θ.

(8)


9.3 Let w be a negative real number, z a 6th root of w.


(a) Show that z (k) = ρ 6


6th roots of w.


cos


( π+2kπ )


+ i sin


( π+2kπ )


, k = 0, 1, 2, 3, 4, 5 is a formula for the





Show all your working.

(8)


(b) Hence determine the 6th roots of −729.

(2)


(c) Given z = cos θ + i sin θ and u + iv = (1 + z)(1 + z2). Prove that v = u tan( 3θ ) and


u2 + v2 = 16 cos2( θ ) cos2(θ)

(10)


[30]

If 4xy−kx2y−5y3 is the imaginary part of the analytic function f(z)=u+iv then, the value of k

(d^2 y)/(dt^2) - 2 dy/dt + 1=0 with y(t=0)=5 and y ̇(t=0)=-9

use laplace transform

Use 9.2 to evaluate sin π

5

,sin 2π

5

and cos π

5

. (9)

10.2 Let z = cosθ + sin θ.

Then z

n = cos nθ + isin θ for all n ∈ N (by de Moivre) and z

−n = cos nθ − isin nθ.

(a) Show that 2 cos nθ = z

n + z

−n

and 2isin nθ = z

n − z

n

. (2)

(b) Show that 2

n

cosn

θ =

Obtain the 6th roots of (-7) , and represent them in an Argand diagram

dy/dt + 4y = 6e ^2t

With y (t=0) = 3, Use laplace transform

F(t)=sint×u(t-2) laplace transform

Construct an entire function f in complex with simple zeros at an = n^(1/2) for n belongs to natural numbers and no other zeros