Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.8 EXERCISES

3.13 Prove thatx^2 m+1−a^2 m+1,wheremis an integer≥1, can be written as

x^2 m+1−a^2 m+1=(x−a)

∏m

r=1

[

x^2 − 2 axcos

(

2 πr

2 m+1

)

+a^2

]

.

3.14 The complex position vectors of two parallel interacting equal fluid vortices
moving with their axes of rotation always perpendicular to thez-plane arez 1
andz 2. The equations governing their motions are
dz∗ 1
dt

=−

i

z 1 −z 2

,

dz∗ 2

dt

=−

i

z 2 −z 1

.

Deduce that (a)z 1 +z 2 ,(b)|z 1 −z 2 |and (c)|z 1 |^2 +|z 2 |^2 are all constant in time,
and hence describe the motion geometrically.
3.15 Solve the equation

z^7 − 4 z^6 +6z^5 − 6 z^4 +6z^3 − 12 z^2 +8z+4=0,
(a) by examining the effect of settingz^3 equal to 2, and then
(b) by factorising and using the binomial expansion of (z+a)^4.
Plot the seven roots of the equation on an Argand plot, exemplifying that complex
roots of a polynomial equation always occur in conjugate pairs if the polynomial
has real coefficients.
3.16 The polynomialf(z) is defined by

f(z)=z^5 − 6 z^4 +15z^3 − 34 z^2 +36z− 48.

(a) Show that the equationf(z) = 0 has roots of the formz=λi,whereλis

real, and hence factorizef(z).

(b) Show further that the cubic factor off(z)canbewrittenintheform

(z+a)^3 +b,whereaandbare real, and hence solve the equationf(z)=0

completely.

3.17 The binomial expansion of (1 +x)n, discussed in chapter 1, can be written for a
positive integernas

(1 +x)n=

∑n

r=0

nCrxr,

wherenCr=n!/[r!(n−r)!].

(a) Use de Moivre’s theorem to show that the sum

S 1 (n)=nC 0 −nC 2 +nC 4 −···+(−1)mnC 2 m,n− 1 ≤ 2 m≤n,

has the value 2n/^2 cos(nπ/4).

(b) Derive a similar result for the sum

S 2 (n)=nC 1 −nC 3 +nC 5 −···+(−1)mnC 2 m+1,n− 1 ≤ 2 m+1≤n,

and verify it for the casesn=6,7and8.

3.18 By considering (1 + expiθ)n, prove that

∑n

r=0

nCrcosrθ=2ncosn(θ/2) cos(nθ/2),

∑n

r=0

nCrsinrθ=2ncosn(θ/2) sin(nθ/2),

wherenCr=n!/[r!(n−r)!].