Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSother exact expressions are possible, e.g. the integral of f(x i + y) over the range0 ≤ y ≤ h, but we will find (27.35) the most useful for our purposes.Although the preceding discussion has implicitly assumed that both of thelimits a and b are finite, with the consequence that N is finite, the general methodcan be adapted to treat some cases in which one of the limits is infinite. It issufficient to consider one infinite limit, as an integral with limits −∞ and ∞ canbe considered as the sum of two integrals, each with one infinite limit.Consider the integralI =∫ ∞af(x) dx,where a is chosen large enough that the integrand is monotonically decreasingfor x>aand falls off more quickly than x −2 . The change of variable t =1/xconverts this integral into∫ 1/a( )1 1I =0 t 2 f dt.tIt is now an integral over a finite range and the methods indicated earlier can beapplied to it. The value of the integrand at the lower end of the t-range is zero.In a similar vein, integrals with an upper limit of ∞ and an integrand that isknown to behave asymptotically as g(x)e −αx ,whereg(x) is a smooth function, canbe converted into an integral over a finite range by setting x = −α −1 ln αt. Again,the lower limit, a, for this part of the integral should be positive and chosenbeyond the last turning point of g(x). The part of the integral for x