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Beta Function

  • Difficulty Level : Medium
  • Last Updated : 16 Jun, 2020

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.

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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_{1}^{0}(1-y)^{p-1}.y^{q-1}(-dy)
    =\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



Example-1:
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
=0.1964



Example-2:
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{1.3.5.7.9}{2.4.6.8.10}.\frac{\pi}{2}
=\frac{63\pi}{256}



Example-3:
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{2.4.6.8}{1.3.5.7.9}
=\frac{384}{945}




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