Beta Function

The Beta function is a unique function and is also called the first kind of Euler’s integrals. The beta function is defined in the domains of real numbers. The notation to represent it is “β”. The beta function is denoted by β(p, q), Where the parameters p and q should be real numbers.

It explains the association between the set of inputs and the outputs. Each input value the beta function is strongly associated with one output value. The beta function plays a major role in many mathematical operations.

Beta function is defined by-
\beta(p, q) = \int_{0}^{1}x^{p-1}(1-x)^{q-1}dx where p>0 and q>0

Some standard results:

  1. Symmetry :
    \beta(p, q) = \beta(q, p)
    \beta(p, q) = \int_{0}^{1}x^{p-1}(1-x)^{q-1}dx
    Put x=1-y
    =\int_{0}^{1}y^{q-1}(1-y)^{p-1}dy = \beta(q, p)
  2. Beta function in terms of trigonometric functions :
    \beta(p, q) = 2\int_{0}^{\pi/2}\sin^{2p-1}x.\cos^{2q-1}xdx
  3. Beta function expressed as improper integral :
    \beta(p, q) = \int_{0}^{\infty} \frac{y^{p-1}}{(1+y)^{p+q}}dy
     = \int_{0}^{\infty} \frac{y^{q-1}}{(1+y)^{p+q}}dy
  4. Relation between beta and gamma functions :
    \beta(p, q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}
  5. \Gamma(p)\Gamma(1-p) = \frac{\pi}{\sin p\pi} where 0<p<1
  6. \int_{0}^{\pi/2}\cos^n x dx = ½ \beta(\frac{1}{2}, \frac{n+1}{2})
  7. \int_{0}^{\pi/2}\sin^n x dx = ½ \beta(\frac{n+1}{2}, \frac{1}{2})
  8. I=\int_{0}^{\pi/2}\sin^p \theta d\theta=\int_{0}^{\pi/2}\cos^p \theta d\theta=

    • \frac{1.3.5….(p-1)}{2.4.6…p}.\frac{\pi}{2} if p is an even positive integer
    • \frac{2.4.6…(p-1)}{1.3.5….p} if p is an odd positive integer
  9. \beta(m, n) = \frac{(m-1)!(n-1)!}{(m+n-1)!} for m, n positive integers

Evaluate \beta(\frac{5}{2}, \frac{3}{2}).

Explanation :
Using result (4) we get,
\beta(\frac{5}{2}, \frac{3}{2})=\frac{\Gamma(5/2)\Gamma(3/2)}{\Gamma(5/2+3/2)}
We know that \Gamma(p+1)=p\Gamma(p)
Thus we get \frac{3/2 \Gamma(\frac{3}{2})\Gamma(\frac{3}{2})}{3!}
=\frac{1}{4}(\frac{1}{2} \Gamma(\frac{1}{2}))^2=\frac{1}{4}\frac{1}{4}\pi

Evaluate \int_{0}^{\pi/2}\sin^{10} \theta  d\theta.

Explanation :
As p=10 is a positive integer, using result (8(i)) we get,
\int_{0}{\pi/2}\sin^{10} \theta d\theta = \frac{}{}.\frac{\pi}{2}

Evaluate \int_{0}^{\pi/2}\cos^9 \theta d\theta.

Explanation :
As p=9 is an odd positive integer, using result 8(ii) we get,
\int_{0}^{\pi/2}\cos^9 \theta d\theta = \frac{}{}

Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using or mail your article to See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :


Please write to us at to report any issue with the above content.