Triple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling

Transcription

1 Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga ( ) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical Logic, Calculus, ad Probability ad Statistics. H was a ispirig Mathmatics profssor who iflucd may of his studts to joi th profssio. Thr hav b a umbr of articls supportig his choic as th bst Mathmatics profssor of th 19 th Ctury (A. Ric, What maks a grat mathmatics tachr? Th Cas of Augustus D Morga, Amr. Math. Mothly 16 (1999), pp ). O of D Morga s sigificat books was his Essay o Probabilitis ad o thir Applicatio to Lif Cotigcis ad Isurac Offics (Logma, Brow, ad Gr, Lodo, 1838). Th itt of th book was to mak Probability Thory accssibl to studts who o mathmatical traiig byod arithmtic. Thus, whil D Morga givs a logical prstatio, h avoids highr lvl mathmatics, ad sticks to a algorithmic, computatioal approach. Bcaus factorials of whol umbrs play a importat rol i Probability Thory, ad wr difficult to comput i th 19 th Ctury for larg whol umbrs, D Morga itroducs th followig algorithm (pp ) to approximat!, whr is a whol umbr. Not that [] i D Morga s otatio mas factorial.

2 I wodrd why his algorithm for! workd. That ld m to a Itrmdiat Algbra/Collg Algbra ivstigatio of th algorithm, which ld to Stirlig s Formula, which approximats!. Havig rctly taught Calculus II ad Calculus III at my collg, I wodrd why Stirlig s Formula so closly approximatd!. That i tur ld to th Gamma Fuctio (sigificatly dvlopd by Eulr), a proof by Mathmatical Iductio (D Morga itroducd th trm Mathmatical Iductio, ad mad th procss rigorous) of th Gamma Fuctio for!, ad a proof of Stirlig s Formula usig doubl itgrals i Polar Coordiats, ad Maclauri Sris. I th procss, I also lard som Mathmatics History ad discovrd som itrstig xrciss for Itrmdiat Algbra, Collg Algbra, Calculus II, ad Calculus III.

3 From D Morga to Stirlig Stirlig s approximatio of! was discovrd by Jams Stirlig ( ), a Scottish Mathmaticia. Stirlig s approximatio stats that!~ Rfrig to D Morga s algorithm for!, ad otig that is a approximatio of log, ad is a approximatio of log, w hav: (1) Tak th log of th umbr, ad subtract : log log = log () Multiply by : log = log (3) Tak log ad add = log + log = log (4) Tak half of th lattr rsult: 1 log log (5) Add th rsults of () ad (4): log + log = log (6) But this a approximatio of th log!, so!~, Stirlig s Formula. This would b a ic xrcis for Itrmdiat Algbra or Collg Algbra studts. Nxt, I wodrd why Stirlig s Formula approximatd!

4 From Stirlig to Eulr ad th Gamma Fuctio Bcaus th Gamma Fuctio givs! for whol umbrs, I xt lookd at th Gamma Fuctio; Eulr sigificatly cotributd to its dvlopmt. Th Gamma Fuctio provids a formula for th factorial of ay complx umbr with a positiv ral part. I my cas, I was itrstd i th factorial of whol umbrs. So, I sought a iductiv proof of th Gamma Fuctio for whol umbrs, a proof that could b usd i Calculus II. Th Gamma Fuctio stats for whol umbrs : x! x dx To iductivly prov that it is tru for whol umbrs: (1) Vrify it is tru for =.! =1 ad () Assum k! = k x x dx (3) Th (k+1)! should qual u = k 1 x ad dv = x dx k1 x x dx w gt x x x dx = dx =1. Usig Itgratio by Parts with k1 x x dx = k1 x x + ( 1) k x k x dx ( k 1) k! ( k 1)! whr k1 x x = by L Hospital s Rul. So, havig faith i th Gamma Fuctio as accuratly productig! for whol umbrs, I watd to prov that Stirlig s formula was a approximatio of th Gamma Fuctio, ad thus a approximatio of!. That ld m to a itrmdiat rsult usig Calculus III ad polar substitutio that I dd bfor I stablishd that Stirlig s formula approximatd th Gamma Fuctio.

5 A Itrmdiat Rsult Bfor I could go us th Maclauri Sris o th Gamma Fuctio, I had to stablish a itrmdiat rsult: x dx = Bcaus f(x) = x is a positiv v fuctio, whr is a whol umbr, Lt I = x dx ad A= 1 x I = dx. Th A = x x dx dx which by Fubii s Thorm ca b rwritt A = x y dx dy = x y dxdy. Usig polar substitutio ( x y r ad dxdy rdrd ) w gt A = r rdrd, whr th limit o is to bcaus w ar i Quadrat 1. r Evaluatig this doubl itgral with u substitutio ( u ), w gt A = or A =. Sic I = A = 4 =. This would b a good xrcis for Calculus III studts.

6 From th Gamma Fuctio (Eulr) to Maclauri (which lads us back to Stirlig) My goal was ow to show that Stirlig s Formula is a approximatio of!, ad thrfor a approximatio of th Gamma Fuctio. To stablish this, I did th followig: x (1)! = x dx by th Gamma Fuctio whr is a whol umbr. () Lt u = (x-) or x= (u+). Also du = dx. Substitutig, w gt (3) u u u u u u 1 du 1 du 1 du u (4) Look at l 1 = l u 1. u (5) Th first two trms of th Maclauri Sris for l 1 = u u u u (6) u = u ad (7) Substitutig th Maclauri Sris approximatio back ito (3) w gt (as ) (8)! ~ u u u u du du = by th Itrmdiat Rsult which taks us back to Stirlig s Formula., This would b a good xrcis for Calculus II studts.

7 A fial ivstigatio As, how closly dos Stirlig s Formula approximat!. O way to look at this is to look at lim. Bcaus of!, it would b impossibl to apply L Hospital s Rul.! Th radr is ivitd to tackl th problm o thir ow. Howvr, usig Scitific Notbook, lim! = 1, idicatig that as icrass, Stirlig s Formula bcoms!. Coclusio I was itrstd i radig D Morga s work o probability (a first ditio I just addd to my atiquaria math book collctio) to gt a historical prspctiv o th subjct. Who would imagi that i so doig, I would d up dvlopig richmt xrciss for classs from Itrmdiat Algbra through Calculus III, ad larig mor Mathmatics History alog th way.