Complex Analysis Answers

Expand f(z)=𝑧+3/𝑧(𝑧2−𝑧−2) in power of z where

a) |𝑧|<1

b) 1<|𝑧|<2

c) |𝑧|>2

Find the value of ∫c 1/𝑍𝑑𝑧, where C is circle 𝑧=𝑒^𝑖∅, 0≤∅≤𝜋

Expand f(z)=𝑧+3/𝑧(𝑧2−𝑧−2) in power of z where

a) |𝑧|<1

b) 1<|𝑧|<2

c) |𝑧|>2

Let F, G be meromorphic functions such that Fⁿ+Gⁿ=1, assuming n≥4 if necessary. Prove that F and G are constants.

(8.1) Let z = z₁/z₂ where z₁ = tan θ + i and z₂ = z₁. Find an expression for z n with n ∈ N.

(8.2) Let z = cos θ − i(1 + sin θ). Determine 2z + i / −1 − iz

Use De Moivre’s Theorem to

(7.1) derive the 4th roots of w = −8i

(7.2) express cos(4θ) and sin(5θ) in terms of powers of cos θ and sin θ

(7.3) expand cos⁶θ in terms of multiple powers of z based on θ

(7.4) express cos³θ sin⁴θ in terms of multiple angles.

Determine for which value (s) of λ the real part of z = 1+λi/1−λi equals zero

Find the roots of the equation:

(5.1) z⁴ + 4 = 0 and z⁴ − 4 = 0

(5.2) Additional Exercises for practice are given below.

Find the roots of

(a) z⁸ − 16 = 0

(b) z⁸ + 16 = 0. 

(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.