Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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

,