Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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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)!].
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