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Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 2

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 40000 30000 20000 10000 5 10 15 20 25 30 Figure 33.2 Plot of the integrand for r 10 We see that the important part of the integrand is the hump centered around x 9. If we find where the integrand of r x has its maximum Ạ e-t tx-1f 0 dx - e-t tx-1 x - 1 e-t tx 2 0 x 1 t 0 t x 1 we see that the maximum varies with x. This could complicate our analysis. To take care of this problem we introduce 1614 the change of variables t xs. r x i e xs xs x 1x ds Jo xx i e-xs sxs 1 ds Jo xx i e-x s-log s s 1 ds o The integrands e x s logs s 1 for r 5 and r 20 are plotted in Figure 33.3. 0.007 0.006 0.005 0.004 0.003 0.002 0.001 2 3 4 Figure 33.3 Plot of the integrand for r 5 and r 20 . We see that the important part of the integrand is the hump that seems to be centered about s 1. Also note that the the hump becomes narrower with increasing x. This makes sense as the e-x s-logs term is the most rapidly varying term. Instead of integrating from zero to infinity we could get a good approximation to the integral by just integrating over some small neighborhood centered at s 1. Since s log s has a minimum at s 1 e-x s-logs has a maximum there. Because the important part of the integrand is the small area around s 1 it makes sense to .