Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

SERIES AND LIMITS


(a) lim
x→ 0

sin 3x
sinhx

, (b) lim
x→ 0

tanx−tanhx
sinhx−x

,


(c) lim
x→ 0

tanx−x
cosx− 1

, (d) lim
x→ 0

(


cosecx
x^3


sinhx
x^5

)


.


4.29 Find the limits of the following functions:


(a)

x^3 +x^2 − 5 x− 2
2 x^3 − 7 x^2 +4x+4

,asx→0,x→∞andx→2;

(b)

sinx−xcoshx
sinhx−x

,asx→0;

(c)

∫π/ 2

x

(


ycosy−siny
y^2

)


dy,asx→0.

4.30 Use Taylor expansions to three terms to find approximations to (a)^4



17, and
(b)^3


26.


4.31 Using a first-order Taylor expansion aboutx=x 0 , show that a better approxi-
mation thanx 0 to the solution of the equation
f(x)=sinx+tanx=2
is given byx=x 0 +δ,where


δ=

2 −f(x 0 )
cosx 0 +sec^2 x 0

.


(a) Use this procedure twice to find the solution off(x) = 2 to six significant
figures, given that it is close tox=0.9.
(b) Use the result in (a) to deduce, to the same degree of accuracy, one solution
of the quartic equation
y^4 − 4 y^3 +4y^2 +4y−4=0.

4.32 Evaluate


lim
x→ 0

[


1


x^3

(


cosecx−

1


x


x
6

)]


.


4.33 In quantum theory, a system of oscillators, each of fundamental frequencyνand
interacting at temperatureT, has an average energyE ̄given by


̄E=

∑∞


n=0nhνe

−nx
∑∞
n=0e

−nx ,

wherex=hν/kT,handkbeing the Planck and Boltzmann constants, respec-
tively. Prove that both series converge, evaluate their sums, and show that at high
temperatures ̄E≈kT, whilst at low temperaturesE ̄≈hνexp(−hν/kT).
4.34 In a very simple model of a crystal, point-like atomic ions are regularly spaced
along an infinite one-dimensional row with spacingR. Alternate ions carry equal
and opposite charges±e. The potential energy of theith ion in the electric field
due to another ion, thejth, is
qiqj
4 π 0 rij


,


whereqi,qjare the charges on the ions andrijis the distance between them.
Write down a series giving the total contributionViof theithiontotheoverall
potential energy. Show that the series converges, and, ifViis written as

Vi=

αe^2
4 π 0 R

,

Free download pdf