Beta Function

The beta function is the name used by Legendre and Whittaker and Watson (1990) for the Eulerian Integral of the Second Kind. To derive the integral representation of the beta function, write the product of two Factorials as

$$m!n! = \int^\infty_0 e^{-u}u^m\,du \int^\infty_0 e^{-v}v^n\,dv.$$ (1)

Now, let $u \equiv x^2$, $v \equiv y^2$, so

$$m!n! = 4 \int_0^\infty e^{-x^2}x^{2m+1}\,dx \int^\infty_0 e^{-y^2}y^{2n+1}\,dy$$

$$= 4\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+y^2)}x^{2m+1}y^{2n+1}\,dx\,dy.$$ (2)

Transforming to Polar Coordinates with $x=r\cos\theta$, $y=r\sin\theta$

$m!n! = 4\int_0^{\pi/2}\!\! \int^\infty_0 e^{-r^2} (r\cos\theta)^{2m+1}(r\sin\theta)^{2n+1} r\,dr\,d\theta$
$= 4\int^\infty_0 e^{-r^2}r^{2m+2n+3}\,dr \int^{\pi/2}_0\cos^{2m+1}\theta\sin^{2n+1}\theta\,d\theta$
$= 2(m+n+1)! \int^{\pi/2}_0 \cos^{2m+1}\theta\sin^{2n+1}\theta\,d\theta.\quad$	(3)

The beta function is then defined by

$$B(m+1,n+1) = B(n+1,m+1) \equiv 2\int^{\pi/2}_0\cos^{2m+1}\theta\sin^{2n+1}\theta\,d\theta = {m!n!\over (m+n+1)!}.$$ (4)

Rewriting the arguments,

$$B(p,q) = {\Gamma(p)\Gamma(q)\over\Gamma(p+q)} = {(p-1)!(q-1)!\over(p+q-1)!}.$$ (5)

The general trigonometric form is

$$\int_0^{\pi/2}\sin^n x\cos^m x\, dx = {\textstyle{1\over 2}}B({\textstyle{1\over 2}}(n+1), {\textstyle{1\over 2}}(m+1)).$$ (6)

Equation (6) can be transformed to an integral over Polynomials by letting $u \equiv\cos^2\theta$,

$$B(m+1,n+1) \equiv {m!n!\over (m+n+1)!} = \int^1_0 u^m(1-u)^n\,du$$ (7)

$$B(m,n) \equiv {\Gamma(m)\Gamma(n)\over \Gamma(m+n)} = \int^1_0 u^{m-1}(1-u)^{n-1}\,du.$$ (8)

To put it in a form which can be used to derive the Legendre Duplication Formula, let $x\equiv \sqrt{u}$, so $u=x^2$ and $du=2x\,dx$, and

$$B(m,n) = \int^1_0 x^{2(m-1)}(1-x^2)^{n-1}(2x\,dx)$$

$$= 2\int_0^1 x^{2m-1}(1-x^2)^{n-1}\,dx.$$ (9)

To put it in a form which can be used to develop integral representations of the Bessel Functions and Hypergeometric Function, let $u \equiv {x/(1+x)}$, so

$$B(m+1,n+1) = \int_0^\infty {u^m\,du\over(1+u)^{m+n+2}}.$$ (10)

Various identities can be derived using the Gauss Multiplication Formula

$$B(np,nq) = {\Gamma(np)\Gamma(nq)\over\Gamma[n(p+q)]} = n^{-n... ...ots B(p+{n-1\over n},q)\over B(q,q)B(2q,q)\cdots B([n-1]q,q)}.$$ (11)

Additional identities include

$$B(p,q+1) = {\Gamma(p)\Gamma(q+1)\over\Gamma(p+q+1)} = {q\over p} {\Gamma(p+1)\Gamma(q)\over\Gamma([p+1]q)}$$

$$= {q\over p}B(p+1,q)$$ (12)

$$B(p,q)=B(p+1,q)+B(p,q+1)$$ (13)

$$B(p,q+1)={q\over p+q}B(p,q).$$ (14)

If $n$ is a Positive Integer, then

$$B(p,n+1)={1\cdot 2\cdots n\over p(p+1)\cdots (p+n)}$$ (15)

$$B(p,p)B(p+{\textstyle{1\over 2}},p+{\textstyle{1\over 2}}) = {\pi\over 2^{4p-1}p}$$ (16)

$$B(p+q)B(p+q,r)=B(q,r)B(q+r,p).$$ (17)

A generalization of the beta function is the incomplete beta function

$B(t;x,y) \equiv \int^t_0 u^{x-1}(1-u)^{y-1}\,du$
$= t^x\left[{{1\over x} + {1-y\over x+1} t + \ldots + {(1-y)\cdots (n-y)\over n!(x+n)} t^n + \ldots}\right].$
	(18)

See also Central Beta Function, Dirichlet Integrals, Gamma Function, Regularized Beta Function

References

Abramowitz, M. and Stegun, C. 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.

Arfken, G. ``The Beta Function.'' §10.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 560-565, 1985.

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.