problem sheet 1

University of NottinghamSchool of Mathematical Sciences

HG1M11 Engineering Mathematics 1

Complex numbers Problem Sheet 1

1. Solve (a) 2x2 − x + 2 = 0, (b) x2 + 2x + 2 = 0.

2. Express the following complex numbers in the form x + iy, where x and y are real.

(a) (3 + 4i)(4− 5i); (b)18− i

2− 3i; (c)

(1 + i)3

(1− i); (d)

(1− i)(1 + 3i)

(2 + i)2.

Show each answer on an Argand diagram.

3. Let

Z =1 + z

1− z�

where z = x + iy and x and y are real. Write Z in the form X + iY where X and Y arereal. That is, find expressions for X and Y in terms of x and y.

4. Let

Z1 = R� Z2 = iωL and Z3 =1

iωC�

where ω, R, L and C are all real constants and let Z be defined by the equation

1

Z=

1

Z1

+1

Z2 + Z3

.

Show that the real part of Z is

Re Z =R

�ωL− 1

ωC

�2

R2 +�ωL− 1

ωC

�2

and find a similar expression for its imaginary part.

5. Express the following in the form reiθ = r(cos θ + i sin θ)� −π < θ ≤ π.

(a) −i (b) 1 + i (c) −8�i√

3− 1�

(d)�1 + i

√3�8

Show each number on the Argand diagram.

6. Complete the following tablex + iy modulus argument polar form exponential form

(a) −2 + 2i√

3

(b) 3 −3π/4

(c) 6(cos π

6+ i sin π

6)

(d) πei

18

7. Find all roots of the following equations.(a) z4 = 1 (b) z4 = 16 (c) z3 = −27i

(d) z3 = −4 (e) z6 = 64(1 + i)/√

2

8. (a) Use de Moivre’s theorem to show that

cos nθ = 1

2(wn + w−n)� (dM)

where w = cos θ + i sin θ.

(b) By writing cos4 θ = ((w + w−1)/2)4, hence confirm that

cos4 θ = 1

8cos 4θ + 1

2cos 2θ + 3

8.

9. The equations for a mechanical vibration problem are found to have the followingmathematical solution

z(t) =eiωt

ω20 − ω2 + iγ

�

where t represents time and ω, ω0 and γ are all positive real physical constants. Althoughz(t) is complex and cannot directly represent a physical solution, it turns out that the realand imaginary parts x(t) and y(t) in z(t) = x(t) + iy(t) can. Polar notation can be usedto extract this physical information efficiently as follows.

(a) Put the numerator in the form

ω2

0− ω2 + iγ = aeiδ�

where you should give explicit expressions for a and δ in terms of ω, ω0 and γ.

(b) Hence find contants b and ϕ such that

x(t) = b cos(ωt + ϕ)

and write a similar expression for y(t).

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