Appendix C Some useful integrals
1 MAXWELL–BOLTZMANN INTEGRALS
To calculate the properties of an MB gas, as in Chapter 6, we need to evaluate the
definiteintegralsoftheform:
IIIn=
∫∞
0
∫∫
ynexp(−by^2 )dy (C.1)
wherenis any positiveinteger. This canbedoneinthree stages.
(i) Equation (C.1) maybe integrated byparts togive
IIIn=[−yn−^1 exp(−by^2 )/ 2 b]∞ 0 +[(n− 1 )/ 2 b]IIIn− 2
Forn≥2, the first term is zero since it vanishes at both limits, giving a simple
recurrence relationbetweenIIInandIIIn− 2 :
IIIn=[(n− 1 )/ 2 b]IIIn− 2 (C.2)
For some purposes, for example the calculation of the rms speed of gas
molecules, the recurrence relation contains enoughinformationbyitself.But
equation (C.2)is usefulin everycase, sinceits application reduces anyintegral
IIInto a known multiple of eitherI 1 orIII 0.
(ii)TheintegralI 1 isobtainedbysimpleintegration, the resultbeing
I 1 = 1 / 2 b (C.3)
(iii) The integralIII 0 takes a little longer to evaluate. A quick method is to considera
two-dimensional problem, to integrate exp(−br^2 )over thewholex−yplane,r
beingthedistancefrom theorigin. Weknowfrom thedefinition (C.1) that
2 III 0 =
∫∞
−∞
∫∫
exp(−bx^2 )dx=
∫∞
−∞
∫∫
exp(−by^2 )dy
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