矢量运算习题.pdf

Assign 5 mathematical Method First name LIANG Last name ZHANG Oct 04 2012 1 15 2 Verify that the sequence n x 0 x 0 satisfi es lim n n x f x dx f 0 where we assume that the function f has derivatives of all orders at x 0 Note Arfken callas this kind of sequence a delta sequence The singulari ty is on the positive side of the origin Solution n x 0 x 0 lim n n x f x dx lim n 0 ne nxf x dx lim n e nxf x 0 0 e nxf x dx lim n e nxf x x 0 e nxf x x 0 e nxf x dx f 0 lim n 0 e nxf x dx f 0 lim n e nx n f x 0 0 e nx n f x dx 1 f 0 lim n e nx n f x x 0 lim n e nx n f x x lim n 0 e nx n f x dx f 0 lim n 0 f x nenx dx f 0 0 f 0 1 15 13 a If we defi ne a sequence n x n 2cosh2nx show that n x dx 1 independent of n 0 b Continuing this analysis show that n x dx 1 2 1 tanhnx un x and lim n un x 0 x 0 Solution a As coshx ex e x 2 sinhx ex e x 2 cosh2x sin2x coshx sinhx coshx sinhx ex e x 1 Besides dcosh x sinh x dx dsinh x cosh x dx 2 1 cosh2x dx cosh2x sinh2x cosh2x dx coshx coshxdx sinhx sinhxdx cosh2x coshxd sinhx sinhxd coshx cosh2x d sinhx coshx n x dx n 2cosh2nx dx 1 2 1 cosh2 nx d nx 1 2d sinhnx coshnx tanhnx 2 tanhnx enx e nx enx e nx n 0 tanhnx x 1 tanhnx x 1 tanhnx 2 1 n x dx 1 3 b x n x dx x n t dt tanhnt 2 x 1 2 tanhnt t x tanhnt t 1 2 tanhnx 1 1 2 1 tanhnx un x for n 0 tanhnx x 1 tanhnx x 1 for x 0 lim n