13.5 HINTS AND ANSWERS
13.17 V ̃(k)∝[− 2 π/(ik)]
∫
{exp[−(μ−ik)r]−exp[−(μ+ik)r]}dr.
13.19 Note that the lower limit in the calculation ofa(z)is0,forz>0, and|z|,for
z<0. Auto-correlationa(z)=[(1/(2λ^3 )] exp(−λ|z|).
13.21 Prove the result fort^1 /^2 by integrating that fort−^1 /^2 by parts.
13.23 (a) Use (13.62) withn=2onL
[√
t]
; (b) use (13.63);
(c) considerL[exp(±at)cosbt]and use the translation property, subsection 13.2.2.
13.25 (a) Note that|lim
∫
g(t)e−stdt|≤|lim∫
g(t)dt|.
(b) (s^2 +as+b) ̄y(s)={c(s^2 +2ω^2 )/[s(s^2 +4ω^2 )]}+(a+s)y(0) +y′(0).
For this damped system, at larget(corresponding tos→0) rates of change
are negligible and the equation reduces toby=ccos^2 ωt. The average value of
cos^2 ωtis^12.
13.27 s−^1 [1−exp(−sa)];ga(x)=xfor 0<x<a,ga(x)=2a−xfora≤x≤ 2 a,
ga(x) = 0 otherwise.