1 6 = 1 6 = + Factorials and Euler s Gamma function


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1 Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a selfcotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio to: i) oiteger argumets, ad ii) egative argumets. Ad as a bous we will obtai a simple expressio for the factorial of very large argumets. You should recall that the terms of Taylor power series expasios ivolve factorials. I the spirit of the extesio of series expasios to icorporate Frobeius series, a similar extesio of the factorial fuctio is useful. The terms of various series expasios are give i terms of oiteger factorials, so we must be able to hadle this. It should also be poited out that the factorial is somewhat aomalous, mathematicallyspeakig, i that it is defied for the itegers but there is othig i betwee ; you certaily ca t differetiate such a fuctio. So you could regard this sectio as providig a smooth iterpolatio of the factorial betwee the discrete iteger poits. I statistical mechaics/thermodyamics you will eed to calculate probabilities ad the umber of ways particles ca be rearraged etc. This ivolves cosiderig factorials of very large umbers (of the order of Avogadro s umber 6 3 ). A mathematical expressio which ca be differetiated ad maipulated is ecessary here. The last part of this sectio will cosider this. The factorial as a itegral We shall defie a ew fuctio ad examie its properties. Previously we have cosidered fuctios defied through the differetial equatios they satisfy. I this case the fuctio will be defied (simply?) as a itegral: 6 x t Fx te. This caot be itegrated (i terms of the elemetary fuctios). It is istructive to itegrate by parts, differetiatig the first term ad itegratig the secod. Thus we are settig x dv t u t, e ad we the have du x t xt, v e. We the use the itegratio by parts rule I d u v I t uv v u t d d d so that x t Fx te x t x t e d t. 6 + PH3 Mathematical Methods
2 Royal Holloway Uiversity of Lodo Departmet of Physics The first term vaishes sice the fuctio is zero at both ed poits. Ad the itegral i the secod term looks suspiciously like the origial defiitio for Fx 6, the oly differece beig that x has bee replaced by x. Thus we coclude Fx xfx. 6 6 What we have here is a recurrece relatio for the fuctio Fx 6; from a give iitial value x we ca thus step up (or dow) oe at a time. Iteger values As a example, we could start from F6ad step up, givig F for all iteger x. Ad fortuately we ca evaluate F6 by direct itegratio, sice t F6 e. So usig the recurrece relatio we build up 6 6 x F F x F6 x 3 F6 3 3F6 3 x 4 F 4 4F :: :::::::::::::::::::::: 6 x F! We have obtaied the factorial fuctio. Thus we coclude that the factorial fuctio may be specified by the itegral x t Fx te 6 ad of course this may be exteded to oiteger argumets. The recurrece relatio also allows the extesio to egative values of x. The Gamma fuctio By covetio mathematicias prefer to use the gamma fuctio (Euler s gamma fuctio) whe extedig the factorial idea to oiteger ad egative argumets. There is a shift of i the defiitio. They use 6 x x! 6 I other words, the gamma fuctio is specified through the itegral x t x6 t e which the has the recurrece relatio x + x x 6 6. PH3 Mathematical Methods
3 Royal Holloway Uiversity of Lodo Departmet of Physics Fractioal argumets Iteger plus a half, or iteger mius a half values of x The itegral expressio for this is / t t e d t, which may be trasformed to the gaussia itegral e y d y π. Thus we have the special value ad we ca build up from this etc. π may be built up from / Negative argumets To treat egative argumets we may use the recurrece relatio to step dow from a give iitial argumet: 6 x x +. 6 x This has the immediate cosequece that 6is ifiite, ad the by extesio the gamma fuctio will be ifiite for all egative iteger x. But for oitegers there is o problem. Thus startig from ½ we would fid, for example, π. The gamma fuctio is plotted i the followig figure. π π 6. PH3 Mathematical Methods 3
4 Royal Holloway Uiversity of Lodo Departmet of Physics (x) x  5 Euler s gamma fuctio Observe the divergeces for egative iteger argumets. Large argumets Stirlig s formula It is easy to approximate the factorial fuctio for large argumet. Sice! , o takig the logarithm we have l 6! l l+ l+ l3+ l l x l x ; we have coverted the product ito a sum. Now whe is large we make oly a small error by covertig this sum ito a itegral. PH3 Mathematical Methods 4
5 Royal Holloway Uiversity of Lodo Departmet of Physics lx x Itegral approximatio to sum The area uder the lower dotted curve is give by approximatio x l x I I l xdx l xd x., so that we adopt the O performig the itegral this gives the approximate expressio l! l. We may take the expoetial of this to fid the approximatio to! itself! exp l These approximate formulae, 6 expl 7 e. l! l ad! e are called Stirlig s approximatio (to the factorial fuctio). e A more systematic aalysis, startig from the itegral expressio for the gamma fuctio ad expadig it term by term gives the series!~ π.... e %&' 88 ()* I applicatios where we are iterested i the logarithm of! whe is large we may igore the square root factor at the left had side ad the iverse series o the right had side; oly the middle part remais, i agreemet with the origial approximatio. Q: Why ca you eglect the square root factor? PH3 Mathematical Methods 5
6 Royal Holloway Uiversity of Lodo Departmet of Physics The importat cocepts of this sectio are: The factorial fuctio is redefied / represeted as a itegral. The recurrece relatio is foud through itegratig by parts. The value for is foud directly by itegratig The itegral expressio allows extesio to oiteger ad egative argumets. 6 6 Euler s Gamma fuctio is defied as x x! Special value / 6 π. Negative argumets accommodated by steppig dow with the recurrece relatio. Stirlig s approximatio for large x obtaied by takig the logarithm of! ad approximatig the resultat sum by a itegral. PH3 Mathematical Methods 6
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