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Chapter 9 Mathematical Preliminaries Stirling s Approximation n n e n n n 2 n log n log k Let I log xdx x log x x 1 n log n n 1 k 1 Fig 9 2 1 n 1 by trapezoid rule 1 1 I log1 log 2 log n 1 log n 2 2 1 n log n n log n 1 log n n n e n n e n 2 Fig 9 2 2 1 1 by midpoint formula I log 2 log 3 log n 1 log n 8 2 7 1 1 n n n log n n 1 log n log n n e n e 8 n 8 2 take antilogs n n n n e take antilogs 7 8 n C where 2 4 e C e 2 7 9 2 Binomial Bounds Show the volume of a sphere of radius n in the n dimensional unit hypercube is n n n V n 1 2nH 2 1 2 n Assuming 0 since the terms are reflected about n 2 the terms grow monotonically and bounding the last by Stirling gives n n n ne n n C n n n n n n n C n n e n n C n n n n n e nn 1 e n C 1 n n n n n n n 1 1 n e ne n n n 1 n 1 n 1 1 C 2 nH 2 C for some C 1 1 1 n 2 n log 1 1 log 1 log 1 H 9 3 n n n 1 termwise by a geometric series bound n n 1 1 n n 1 3 2 1 with ratios the 1 n n 1 n n 2 k n 2 n 1 n 1 1 actual ratios between te rms So 1 2 k 0 1 1 1 k n n C C 1 nH 2 nH 2 nH 2 2 2 2 k n k 0 1 k 0 n 1 2 1 as n n 2 N b n n n 1 n nH 2 1 n 2 2 2 2 k 0 k k 0 k 9 3 The Gamma Function Idea extend the factorial to non integral arguments by convention Let n e x x n 1dx 0 For n 1 integrate by parts dg e xdx f xn 1 n x n 1 x e n 1 e x x n 2 dx 10 0 n n 1 n 1 1 e dx e 0 2 1 3 2 x x 0 1 4 3 n n 1 9 4 1 2 x t 2 1 x t2 1 e x dx e 2tdt t 2 0 0 2 t2 2 e dt e dt t called the error integral 0 1 1 x y x2 y2 e dx e dy e dxdy 2 2 dr dx dy rd r area rdrd 2