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 self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio to: i) o-iteger 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 o-iteger factorials, so we must be able to hadle this. It should also be poited out that the factorial is somewhat aomalous, mathematically-speakig, 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 o-iteger 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 o-iteger 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 o-itegers 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 re-defied / 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 o-iteger 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