Factorial, Gamma and Beta Functions

1 Factorial, Gamma and Beta Gamma Beta Franois Antoine Arbogast (1759 - 1803) a French mathematician, is generally creditedwith being the first to introduce the concept of thefactorialas a product of a fixed numberof terms in arithmetic progression. In an effort to generalize the factorial function to non-integer values, theGamma functionwaslater presented in its traditional integral form bySwiss mathematician Leonhard Euler (1707-1783). In fact, the integral form of the Gammafunction is referred to as the second Eulerian integral. Later, because of its great importance,it was studied by other eminent mathematicians like Adrien-Marie Legendre (1752-1833),Carl Friedrich Gauss (1777-1855), Cristoph Gudermann (1798-1852), Joseph Liouville (1809-1882), Karl Weierstrass (1815-1897), Charles Hermite (1822 - 1901), as well as many first reported use of the Gamma symbol for this function was by Legendre in first Eulerian integral was introduced by Euler and is typically referred to by its morecommon name, theBeta use of the Beta symbol for this function was firstused in 1839 by Jacques Binet (1786 - 1856).

2 Atthe same time as Legendre and Gauss, Cristian Kramp (1760 - 1826) worked on thegeneralized factorial function as it applied to non-integers. His work on factorials was in-dependent to that of Stirling, although Sterling often receives credit for this effort. He didachieve one first in that he was the first to use the notation n! although he seems not toberemembered today for this widely used mathematical historical perspective of the Gamma function is given in the work of Godefroy4as well as other associated authors given in the references at the end of this , , p. 2713 Elements d arithmtique universelle , 18084M. Godefroy,Lafonction Gamma ; Theorie, Histoire, Bibliographie,Gauthier-Villars, Paris (1901)2 Definitions1. Factorialn!=n(n 1)(n 2)..3 2 1for all integers,n>02. Gammaalso known as:generalized factorial, Euler s second integralThe factorial function can be extended to include all real valued argumentsexcluding the negative integers as follows:z!= 0e ttzdtz = 1, 2, 3.

3 Or as the Gamma function: (z)= 0e ttz 1dt=(z 1)!z = 1, 2, 3,..3. Digammaalso known as:psi function, logarithmic derivative of the Gamma function (z)=dln (z)dz= (z) (z)z = 1, 2, 3,..4. Incomplete GammaThe Gamma function can be written in terms of two components as follows: (z)= (z, x)+ (z, x)where the incomplete Gamma function, (z, x),isgiven as3 (z, x)= x0e ttz 1dtx >0and its compliment, (z, x),as (z, x)= xe ttz 1dtx >05. Betaalso known as:Euler s first integralB(y, z)= 10ty 1(1 t)z 1dt= (y) (z) (y+z)6. Incomplete BetaBx(y, z)= x0ty 1(1 t)z 1dt0 x 1and the regularized (normalized) form of the incomplete Beta functionIx(y, z)=Bx(y, z)B(y, z)4 TheoryFactorial FunctionThe classical case of the integer form of the factorial function,n!,consists of the product ofnand all integers less thann,downto1,asfollowsn!= n(n 1)(n 2)..3 2 1n=1,2,3,..1n=0( )where by definition,0! = integer form of the factorial function can be considered as a special case of two widelyused Functions for computing factorials of non-integer arguments, namely thePochham-mer s polynomial,given as(z)n= z(z+1)(z+2).

4 (z+n 1) = (z+n) (z)n>0=(z+n 1)!(z 1)!1=0!n=0( )and thegamma function(Euler s integral of the second kind). (z)=(z 1)!( )While it is relatively easy to compute the factorial function for small integers, it is easy to seehow manually computing the factorial of larger numbers can be very tedious. Fortunatelygiven the recursive nature of the factorial function, it is very well suited to a computer andcan be easily programmed into a function or subroutine. The two most common methodsused to compute the integer form of the factorial aredirect computation:use iteration to produce the product of all of the counting numbersbetweennand1,asinEq. computation:define a function in terms of itself, where values of the factorialare stored and simply multiplied by the next integer value in the sequence5 Another form of the factorial function is the double factorial, defined asn!! = n(n 2)..5 3 1n>0oddn(n 2)..6 4 2n>0even1n= 1,0( )The first few values of the double factorial are given as0!

5 ! = 15!! = 151!! = 16!! = 482!! = 27!! = 1053!! = 38!! = 3844!! = 89!! = 945 While there are several identities linking the factorial function to the double factorial, perhapsthe most convenient isn!=n!!(n 1)!!( )Potential and Combinations:The combinatory functionC(n, k)(nchoosek)allows a concise statement of the Binomial Theorem using symbolic notation and inturn allows one to determine the number of ways to choosekitems fromnitems,regardless of combinatory function provides the binomial coefficients and can be defined asC(n, k)=n!k!(n k)!( )It has uses in modeling of noise, the estimation of reliability in complex systems aswell as many other engineering FunctionThe factorial function can be extended to include non-integer arguments through the use ofEuler s second integral given asz!= 0e ttzdt( )Equation is often referred to as thegeneralized factorial a simple translation of thez variable we can obtain the familiargamma functionas follows (z)= 0e ttz 1dt=(z 1)!

6 ( )The Gamma function is one of the most widely used special Functions encountered in advancedmathematics because it appears in almost every integral or series representation of otheradvanced mathematical s first establish a direct relationship between the Gamma function given in Eq. andthe integer form of the factorial function given in Eq. Given the Gamma function (z+1)=z!use integration by parts as follows: udv=uv vduwhere from Eq. we seeu=tz du=ztz 1dtdv=e tdt v= e twhich leads to (z+1)= 0e ttzdt= e ttz 0+z 0e ttz 1dt7 Given the restriction ofz>0for the integer form of the factorial function, it can be seenthat the first term in the above expression goes to zero since, whent=0 tn 0t= e t 0 Therefore (z+1)=z 0e ttz 1dt (z)=z (z),z>0( )Whenz=1 tz 1=t0=1,and (1) = 0! = 0e tdt= e t 0=1and in turn (2) = 1 (1) = 1 1=1! (3) = 2 (2) = 2 1=2! (4) = 3 (3) = 3 2=3!In general we can write (n+1)=n!n=1,2,3,..( )8 The Gamma function constitutes an essential extension of the idea of a factorial, since theargumentzis not restricted to positive integer values, but can vary Eq.

7 , the Gamma function can be written as (z)= (z+1)zFrom the above expression it is easy to see that whenz=0,the Gamma function approaches or in other words (0)is the recursive nature of the Gamma function, it is readily apparent that the gammafunction approaches a singularity at each negative , for all other values ofz, (z)is defined and the use of the recurrence relationshipfor factorials, (z+1)=z (z)effectively removes the restriction thatxbepositive, which the integral definition of thefactorial requires. Therefore, (z)= (z+1)z,z =0, 1, 2, 3,..( )Aplot of (z)is shown in Figure other definitions of the -function are available that can be attributed to the pio-neering mathematicians in this areaGauss (z)=limn n!nzz(z+1)(z+2)..(z+n),z =0, 1, 2, 3,..( )Weierstrass1 (z)=ze z n=1 1+zn e z/n( )9 4 2024z 15 10 5051015 z Figure : Plot of Gamma Functionwhere is theEuler-Mascheroni constant,defined by =limn n k=11k ln(n)= 56649 ( )An excellent approximation of is given by the very simple formula =12 3 10 1 = forms of the Gamma function are obtained through a simple change of variables, asfollows (z)=2 0y2z 1e y2dybylettingt=y2( ) (z)= 10 ln1y z 1dybylettinge t=y( )10 Relations Satisfied by the -FunctionRecurrence Formula (z+1)=z (z)( )Duplication Formula22z 1 (z) z+12 = (2z)( )Reflection Formula (z) (1 z)= sin z( )Some Special Values of the Gamma FunctionUsing Eq.

8 Or Eq. we have (1/2)=( 1/2)!=2 0e y2dy I= ( )where the solution toIis obtained from Schaum s Handbook of Mathematical Functions (Eq. ).11 Combining the results of Eq. with the recurrence formula, we see (1/2) = (3/2) =12 (1/2) = 2 (5/2) =32 (3/2) =32 2=3 n+12 =1 3 5 (2n 1)2n n=1,2,3,..Forz>0, (z)has a single minimum within the range1 z 21450where (z)= selected 10 decimal place values of (z)are found inTable : 10 Decimal Place Values of (z)for1 z 2z (z) 00000 Forother values ofz(z =0, 1, ), (z)can be computed by means of therecurrence Representation of the Factorial and Gamma FunctionsAsymptotic expansions of the factorial and Gamma Functions have been developed forz>> expansion for the factorial function isz!= (z+1)= 2 z zze zA(z)( )whereA(z)=1+112z+1288z2 13951840z3 5712488320z4+ ( )The expansion for the natural logarithm of the Gamma function isln (z)= z 12 lnz z+12ln(2 )+112z 1360z3+11260z5 11680z7+ ( )The absolute value of the error is less than the absolute value of the first term values ofz, asz ,both expansions lead toStirling s Formula,given asz!

9 = 2 zz+1/2e z( )Even though the asymptotic expansions in Eqs. and were developed for very largevalues ofz,they give remarkably accurate values ofz!and (z)for small values shows the relative error between the asymptotic expansion and known accurate valuesfor arguments between1 z 7,where the relative error is defined asrelative error =approximate value accurate valueaccurate value13 Table : Comparison of Approximate value ofz!byEq. and (z)byEq. withthe Accurate values of Mathematica ! !Mathematicaerror (z) (z) 9469 10 2549 10 8981 10 8900 10 7005 10 9965 10 9267 10 9997 10 9756 10 9999 10 9901 10 9999 10 9954 10 9999 10 11 The asymptotic expansion for (z)converges very quickly to give accurate values for rela-tively small values asymptotic expansion forz!converges less quickly and doesnot yield 9 decimal place accuracy even whenz= accurate values of (z)for smallzcan be obtained by means of the recurrence , if we want (1+z)where0 z 1,then by means of the recurrence formulawecan write (1 +z)= (n+z)(1 +z)(2 +z)(3 +z).

10 (n 1+z)( )wherenis an integer greater ,wehave (1 + ) = ( )( )( )( )( )= 0699 This value can be compared with the10decimal place value given previously in Table that the absolute error is approximately3 10 accuracy canbeobtained by means of the above equation withn=6and0 z Approximation of (z+1)within 0 z 1 Numerous polynomial approximations which are based upon the use of Chebyshev polyno-mials and the minimization of the maximum absolute error have been developed for varyingdegrees of accuracy. One such approximation developed for0 z 1due to Hastings8is (z+1) =z!=1+z(a1+z(a2+z(a3+z(a4+z(a5+z(a6+z(a7 +a8z))))))) + (z)( )where| (z)| 3 10 7and the coefficients in the polynomial are given asTable : Coefficients of Polynomial of Eq. 1652a5= 4078a2= 5891a6= 9394a3= 6937a7= 7818a4= 6857a8= 834315 Series Expansion of 1/ (z)for|z| The function1/ (z)is an entire function defined for all values be expressed asaseries expansion according to the relationship1 (z)= k=1 Ckzk,|z| ( )where the coefficientsCkfor0 k 26,accurate to16decimal places are tabulated inAbramowitz and place accuracy one can write1 (z)=19 k=1 Ckzk( )where the coefficients are listed belowTable : Coefficients of Expansion of1/ (z)of Eq.