Question

B(2,3)
The value of beta function is equal to​

Answer 1

Answer:

Explanation: \beta(3, 2) = \frac{\Gamma(3).\Gamma(2)}{\Gamma(3+2)}

= \frac{2!1!}{4!} = \frac{1}{12}.

Answer 2

$\displaystyle \sf{B(2,3) = \frac{1 }{12} }$

Given :

The beta function B(2,3)

To find :

The value of function

Solution :

Step 1 of 3 :

Write down the given function

The beta function is B(2,3)

Step 2 of 3 :

Express in terms of Gamma function

$\displaystyle \sf{ B(x,y) = \frac{ \Gamma (x)\Gamma (y)}{\Gamma (x + y)} }$

Thus we get

$\displaystyle \sf{ B(2,3) = \frac{ \Gamma (2)\Gamma (3)}{\Gamma (2 + 3)} }$

$\displaystyle \sf{ \implies B(2,3) = \frac{ \Gamma (2)\Gamma (3)}{\Gamma (5)} }$

Step 3 of 3 :

Find the value of the function

We use the formula

$\sf \Gamma (n + 1) =n!$

Thus we get

$\displaystyle \sf{ B(2,3) }$

$\displaystyle \sf{ = \frac{ \Gamma (2)\Gamma (3)}{\Gamma (5)} }$

$\displaystyle \sf{ = \frac{1! \: 2! }{4!} }$

$\displaystyle \sf{ = \frac{1 \times 2}{4 \times 3 \times 2} }$

$\displaystyle \sf{ = \frac{1 }{12} }$

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