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Beta Function

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BetaFunction

The beta function B(p,q) is the name used by Legendre and Whittaker and Watson (1990) for the beta integral (also called the Eulerian integral of the first kind). It is defined by

B(p,q)=(Gamma(p)Gamma(q))/(Gamma(p+q))=((p-1)!(q-1)!)/((p+q-1)!).
(1)

The beta function B(a,b) is implemented in the Wolfram Language as Beta[a, b].

To derive the integral representation of the beta function, write the product of two factorials as

m!n!=int_0^inftye^(-u)u^mduint_0^inftye^(-v)v^ndv.
(2)

Now, let u=x^2, v=y^2, so

m!n!=4int_0^inftye^(-x^2)x^(2m+1)dxint_0^inftye^(-y^2)y^(2n+1)dy
(3)
=int_(-infty)^inftyint_(-infty)^inftye^(-(x^2+y^2))|x|^(2m+1)|y|^(2n+1)dxdy.
(4)

Transforming to polar coordinates with x=rcostheta, y=rsintheta

m!n!=int_0^(2pi)int_0^inftye^(-r^2)|rcostheta|^(2m+1)|rsintheta|^(2n+1)rdrdtheta
(5)
=int_0^inftye^(-r^2)r^(2m+2n+3)drint_0^(2pi)|cos^(2m+1)thetasin^(2n+1)theta|dtheta
(6)
=4int_0^inftye^(-r^2)r^(2m+2n+3)drint_0^(pi/2)cos^(2m+1)thetasin^(2n+1)thetadtheta
(7)
=2(m+n+1)!int_0^(pi/2)cos^(2m+1)thetasin^(2n+1)thetadtheta.
(8)

The beta function is then defined by

B(m+1,n+1)=2int_0^(pi/2)cos^(2m+1)thetasin^(2n+1)thetadtheta
(9)
=(m!n!)/((m+n+1)!).
(10)

Rewriting the arguments then gives the usual form for the beta function,

B(p,q)=(Gamma(p)Gamma(q))/(Gamma(p+q))
(11)
=((p-1)!(q-1)!)/((p+q-1)!).
(12)

By symmetry,

B(p,q)=B(q,p).
(13)

The general trigonometric form is

int_0^(pi/2)sin^nxcos^mxdx=1/2B(1/2(n+1),1/2(m+1)).
(14)

Equation (14) can be transformed to an integral over polynomials by letting u=cos^2theta,

B(m+1,n+1)=(m!n!)/((m+n+1)!)
(15)
=int_0^1u^m(1-u)^ndu
(16)
B(m,n)=(Gamma(m)Gamma(n))/(Gamma(m+n))
(17)
=int_0^1u^(m-1)(1-u)^(n-1)du.
(18)

For any z_1,z_2 with Re[z_1],Re[z_2]>0,

B(z_1,z_2)=B(z_2,z_1)
(19)

(Krantz 1999, p. 158).

To put it in a form which can be used to derive the Legendre duplication formula, let x=sqrt(u), so u=x^2 and du=2xdx, and

B(m,n)=int_0^1x^(2(m-1))(1-x^2)^(n-1)(2xdx)
(20)
=2int_0^1x^(2m-1)(1-x^2)^(n-1)dx.
(21)

To put it in a form which can be used to develop integral representations of the Bessel functions and hypergeometric function, let u=x^2/(1-x^2), so

B(m+1,n+1)=int_0^infty(u^mdu)/((1+u)^(m+n+2)).
(22)

In particular, the substitution u=x^p and the reflection relation give, for p>1,

int_0^infty(dx)/(1+x^p)=1/pB(1/p,1-1/p)
(23)
=pi/pcsc(pi/p).
(24)

Derivatives of the beta function are given by

d/(da)B(a,b)=B(a,b)[psi_0(a)-psi_0(a+b)]
(25)
d/(db)B(a,b)=B(a,b)[psi_0(b)-psi_0(a+b)]
(26)
(d^2)/(db^2)B(a,b)=B(a,b){[psi_0(b)-psi_0(a+b)]^2+psi_1(b)-psi_1(a+b)},
(27)
(d^2)/(dadb)B(a,b)=B(a,b){[psi_0(a)-psi_0(a+b)]×[psi_0(b)-psi_0(a+b)]-psi_1(a+b)},
(28)

where psi_n(x) is the polygamma function.

Various identities can be derived using the Gauss multiplication formula

B(np,nq)=(Gamma(np)Gamma(nq))/(Gamma(n(p+q)))
(29)
=n^(-nq)(B(p,q)B(p+1/n,q)...B(p+(n-1)/n,q))/(B(q,q)B(2q,q)...B((n-1)q,q)).
(30)

Additional identities include

B(p,q+1)=(Gamma(p)Gamma(q+1))/(Gamma(p+q+1))
(31)
=q/p(Gamma(p+1)Gamma(q))/(Gamma((p+1)+q))
(32)
=q/pB(p+1,q)
(33)
B(p,q)=B(p+1,q)+B(p,q+1)
(34)
B(p,q+1)=q/(p+q)B(p,q).
(35)

If n is a positive integer, then

B(p,n+1)=(1·2...n)/(p(p+1)...(p+n)).
(36)

In addition,

B(p,p)B(p+1/2,p+1/2)=pi/(2^(4p-1)p)
(37)
B(p,q)B(p+q,r)=B(q,r)B(q+r,p).
(38)

The beta function is also given by the product

B(x,y)=(x+y)/(xy)product_(k=1)^infty(1+(x+y)/k)/((1+x/k)(1+y/k))
(39)

(Andrews et al. 1999, p. 8).

Gosper gave the general formulas

product_(i=0)^(2n)B(i/(2n+1)+a,i/(2n+1)+b) =((2n+1)^((2n+1)/2)pi^nB(n,1/2[(b+a)(2n+1)+1])B(a(2n+1),b(2n+1)))/((n-1)!)
(40)

for odd n, and

product_(i=0)^(2n-1)B(i/(2n)+a,i/(2n)+b) =(n^npi^nB(n,2(a+b)n)B(2an,2bn))/(2^(2(a+b)n-n-1)(n-1)!B((a+b)n,(a+b+1)n)),
(41)

which are an immediate consequence of the analogous identities for gamma functions. Plugging n=1 and n=2 into the above give the special cases

B(a,b)B(a+1/3,b+1/3)B(a+2/3,b+2/3)=(6pisqrt(3)B(3a,3b))/(1+3(a+b))
(42)
B(a,b)B(a+1/4,b+1/4)B(a+1/2,b+1/2)B(a+3/4,b+3/4)=(2^(3-4(a+b))pi^2B(4a,4b))/((a+b)[1+4(a+b)]B(2(a+b),2(a+b+1))).
(43)

See also

Beta Integral, Central Beta Function, Dirichlet Beta Function, Dirichlet Integrals, Gamma Function, Incomplete Beta Function, Regularized Beta Function

Related Wolfram sites

https://functions.wolfram.com/GammaBetaErf/Beta/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Beta Function" and "Incomplete Beta Function." §6.2 and 6.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 258 and 263, 1972.Andrews, G. E.; Askey, R.; and Roy, R. Special Functions. Cambridge, England: Cambridge University Press, 1999.Arfken, G. "The Beta Function." §10.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 560-565, 1985.Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Beta Function." §1.5 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 9-13, 1981.Jeffreys, H. and Jeffreys, B. S. "The Beta Function." §15.02 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 463-464, 1988.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 6-9, 1998.Krantz, S. G. "The Beta Function." §13.1.11 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 157-158, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 425, 1953.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gamma Function, Beta Function, Factorials, Binomial Coefficients" and "Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206-209 and 219-223, 1992.Spanier, J. and Oldham, K. B. "The Incomplete Beta Function B(nu;mu;x)." Ch. 58 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 573-580, 1987.Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Beta Function

Cite this as:

Weisstein, Eric W. "Beta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BetaFunction.html

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