N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio oe has he coiuous gamma fucio Γ(). Boh are defied by he same iegral Oe iegraio by pars yields- ( )! 0 exp( ) d from which follow he ideiies- 0 0 exp( ) d exp( ) d (+)!=!(+) ad Γ(+)=Γ() We also have ha 0!=!= ad Γ(0)=, Γ(/)=sqr(π) ad Γ()=. A plo of! ad Γ(+) follow-
The blue dos are! values while he red curve represes he coiuous Γ(+) fucio. The curve reaches a miimum value of Γ()= 0.88560 a =.46. Oe ypically fids he values for Γ() are abulaed oly i he rage <<, sice he res ca be quickly geeraed via he above recurrece formula. Alhough here is o rue facorial for egaive iegers, oe ca exed he Γ() fucio o egaive ad obai o ifiie values whe is a egaive o-ieger. We ca also express he derivaive of he gamma fucio as he iegral- d( ) d 0 exp( ) d ( ) d d 0 l( ) exp( ) d This derivaive goes oward - as ->0, has zero value ear =.46 ad akes o progressively larger posiive values as heads oward plus ifiiy. Noice ha his las iegral is jus he Laplace rasform of (-) l() afer s is se equal o uiy. A fucio relaed o Γ(x) is he digamma fucio- ( x) d( x) / dx ( x) d[l ( x)] dx I has he value ψ()=-γ where γ=0.57756649.. is he Euler cosa. Oe ca also sum he reciprocals of various combiaios of!. We have, amog may oher examples, ha- exp() I () 0 0 cosh() sih() 0! (!) 0 0.78888459045...795853033607...543080634854.. ()!.7509364380.. ( ) Noice also ha- ()!=[ 3 5 (-)][ 4 6 ]= [ 3 5 (-)]! Thus we have ha- 3 5 (-)=()!/(!)
From his i follows ha he produc of he firs five odd umbers equals 0!/(3*5!)=945. This las form also allows oe o wrie cerai ifiie series i compac form. For example, we have ha- x x ( 3) x ( 35) x... (m)! 4 6 3 m!! 3! m 0 ( m! ) As we leared i our earlier discussios o Legedre polyomials P (x), hese ca be geeraed by he geeraig fucio - P ( x) x 0 So o seig ε=x- we ca wrie- x m + 0 P ( x) (m)! m! [ (x )] m m0 ( m!) (!) 3 x [(3x ) / ] O( ) which produces he Legedre polyomials. 4! [ (x )] 4 (!) Oe ca also use he gamma fucio Γ() o evaluae 3 5 -. We have Γ(/)=sqr(π) so ha Γ(3/)=sqr(π)/, Γ(5/)=( 3)sqr(π)/ ad Γ(7/)=( 3 5)sqr(π)/ 3. From his i follows ha- [ 3 5 (-)]= Γ(+/)/sqr(π) [ (x )] Combiig his resul wih he form of ()! give earlier, oe obais he Legedre Duplicaio Formula-! ()! ( ) Tryig his ou for =5, we fid- 0!=( 0 5!)Γ(/)/sqr(π)=368800... Aoher combiaio of facorials which ofe arises is he famous biomial coefficie- C m =!/[(m!(-m)!] I is produced by he followig biomial expasio-
( a b) a a b /! ( ) a b /! m0 C m a m b m Noe ha he C m for a fixed jus represes he umbers i he h row of a Pascal riagle. Thus he 4 h row has he coefficies C 4m =4!/[(m!(4-m)!] which are -4-6-4-. Aoher exesio of he facorial is he produc of squares which read- F()= 4 9 6 This is easy o evaluae by oig F() is jus he produc of! wih iself. Tha is- F()=( 3 )( 3 )=(!) I also follows ha he produc of he firs ph powers of he iegers equals (!) p. Thus 8 7 64 5=(5!) 3 =78000 Nex we examie he value of Γ(+/). Usig he Legedre Duplicaio Formula ad he form for ()! give earlier, we have- ()! ( )! This allows oe o fid he half-ieger gamma fucio. I says Γ(3/)=30!sqr(π)/( 30 5!)= (6908335369375/3768)sqr(π) As expeced his value lies bewee 4! Ad 5!. I is also possible o develop gamma fucio ideiies o foud i exisig mahemaical hadbooks. Oe of hese is- G()=Γ(+/) Γ(-/) We develop he geeral value for G() by sarig wih = where G()=π/. Nex a = we have G()=+3π/8 ad a =3 we fid G(3)=45π/3. This suggess ha ( ) ( ) G( ) { 3 5.. ( 3) } [ ( )] This ideiy checks for all values of ried for of oe or greaer.
Aoher variaio is he gamma produc fucio- P(x,y)=Γ(x+y) Γ(x-y) which reduces o (x+y-)! (x-y-)! whe x ad y are iegers. A coour plo of his fucio for x>0 ad -4<y<4 looks like his- The coours form closed curves ad P(x,y) goes o ifiiy whe y=±(x+) sice gamma for ay egaive ieger i ubouded. Alerae srip regios bewee he ui slope curves also show fiie valued coours. The miimum coour value occurs ear x=.46 ad y=0 ad has he value P=0.7844. Fially we look a he gamma fucio Γ(z) whe z=x+iy is a complex umber. Here he bes approach is o use he iegral defiiio- ( x iy) 0 xiy exp( ) d 0 x exp( ) cos( y l ) i si( y l ) d From he iegral we see ha Γ(x+iy) has a real ad imagiary par represeed by wo differe iegrals. We fid- Γ(+i)= 0.49805668-i0.54949883 so ha Γ(+i) Γ(-i)= Γ(+i) =0.7090550... Also we have ha-
0 cos[l( )]exp( )) d 0.49805668.. We ca also plo Γ(x+iy)=u+iv i he u-v plae. This ca produce some ieresig figures such as he followig- I he firs we plo Γ(z) for z=x+i o ge a ru-away spiral paer. For he secod figure we have se z=+iy. I produces a closed double loop. May oher plos are possible by jus seig x or y o differe cosa values. March 03