Complex Analysis Answers

(4.1) Determine the complex numbers i²⁶⁶⁶ and i¹⁴⁵.

(4.2) Let z₁ = (6) −i −1+i and z₂ = 1+i 1−i . Express z₁z₃/z2 , z₁z₂/z3 , and z₁/z₃z₂ in both polar and standard forms.

(4.3) Additional Exercises for practice: Express z₁ = −i, z₂ = −1 − i √ 3, and z₃= − √ 3 + i in polar form and use your results to find z⁴₃ /z²₁ z^-1₂. Find the roots of the polynomials below.

(a) P(z) = z² + a for a > 0

(b) P(z) = z³ − z² + z − 1.

(c) Find the roots of z (4) 3 − 1

(d) Find in standard forms, the cube roots of 8 − 8i

(e) Let w = 1 + i. Solve for the complex number z from the equation z⁴ = w³ . (4.4) Find the value(s) for λ so that α = i is a root of P(z) = z² + λz − 6.

Let θ be a real number. Then check whether the matrices 

cos θ − sin θ

sin θ cos θ

and 

e

iθ −0

0 e

−iθ 

are similar over the field of complex numbers.

Use de moivres theorem to

1.derive the 4th roots of w=-8i.

2.express cos(4@) and sin(5@) in terms of powers of cos@ and sin@.

3.expand cos^6@ in terms of multiple powers of z based on @.

4.express cos^3@sin^4@ in terms of multiple angles.

NOTE:@ represents theta.

Interpret each of the following transformation in the complex plane

(i) T1:z→ w, given by w=-z* where z* is the conjugate of z.

(ii) T2:z→w given by w=3z-1+2i.

Find the invariant complex number under the transformation T2.

1. Determine the poles and residue at each poles of

f(z)=2z+1/z²-z-2 over C=|z|=5/2 Hence Evaluate

∮c 2z+1/z²-z-2 dz over C=|z|=5/2

Given that 𝑧1 = 3 + 𝑖 𝑎𝑛𝑑 𝑧2 = 2 − 𝑖: i. Find the modulus and argument of 𝑧1 𝑧2 (5 marks) ii. Express 𝑧1 𝑧2 in polar and exponential form iii. Use de Moivre’s theorem to find an expression for ( 𝑧1 𝑧2 ) 4 

1. Evaluate the integral: ∫c 1/z² dz

Where the contour C is

a) The line segment with initial point -1 and final point i.

b)The arc of the unit circle Imz>=0 with initial point -1 and final point i.

2. Evaluate the complex number :

[(15+7j)(3-2j)*/(4+6j)*(3∠60°)]*

3. Determine whether the seris is convergent or divergent. If it is convergent find its sum

Σ upper limit ∞, lower limit k=3 (8^-k 4^k+2 -3^k+3/6^k)

Evaluate integral of (z-3)^4 where c is the circle |z-3|=4

If z + 1/z=2 cos theta , theta belongs to R.  Show that |z|=1 and for any n belongs to Z, z^n + 1/z^n  =2 cos n theta.