Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSsin 3x(a) limx→0 sinh x ,tan x − x(c) limx→0 cos x − 1 ,tan x − tanh x(b) limx→0 sinh x − x ,( cosec x(d) lim − sinh x ).x→0 x 3 x 54.29 Find the limits of the following functions:x 3 + x 2 − 5x − 2(a), as x → 0, x →∞and x → 2;2x 3 − 7x 2 +4x +4sin x − x cosh x(b), as x → 0;sinh x − x∫ π/2( )y cos y − sin y(c)dy, as x → 0.x y 24.30 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 about x = x 0 , show that a better approximationthan x 0 to the solution of the equationf(x) =sinx +tanx =2is given by x = x 0 + δ, where2 − f(x 0 )δ =.cos x 0 +sec 2 x 0(a) Use this procedure twice to find the solution of f(x) = 2 to six significantfigures, given that it is close to x =0.9.(b) Use the result in (a) to deduce, to the same degree of accuracy, one solutionof the quartic equationy 4 − 4y 3 +4y 2 +4y − 4=0.4.32 Evaluate[ ( 1lim cosec x − 1x→0 x 3 x − x )].64.33 In quantum theory, a system of oscillators, each of fundamental frequency ν andinteracting at temperature T , has an average energy Ē given by∑ ∞n=0Ē = ∑ nhνe−nx∞ ,n=0 e−nxwhere x = hν/kT , h and k being the Planck and Boltzmann constants, respectively.Prove that both series converge, evaluate their sums, and show that at hightemperatures Ē ≈ kT, whilst at low temperatures Ē ≈ hν exp(−hν/kT ).4.34 In a very simple model of a crystal, point-like atomic ions are regularly spacedalong an infinite one-dimensional row with spacing R. Alternate ions carry equaland opposite charges ±e. The potential energy of the ith ion in the electric fielddue to another ion, the jth, isq i q j,4πɛ 0 r ijwhere q i , q j are the charges on the ions and r ij is the distance between them.Write down a series giving the total contribution V i of the ithiontotheoverallpotential energy. Show that the series converges, and, if V i is written asV i =148αe24πɛ 0 R ,