Question

If $\beta(m, n)$, $\Gamma(m)$ and $\Gamma(n)$ for $m > 0, n > 0$ represents beta and gamma functions respectively, then which one of the following is correct?

• A. $\beta(m, n) = \frac{\Gamma(m) + \Gamma(n)}{\Gamma(m+n)}$
• B. $\beta(m, n) = \frac{\Gamma(m+n)}{\Gamma(m) + \Gamma(n)}$
• C. $\beta(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}$
• D. $\beta(m, n) = \frac{\Gamma(m+n)}{\Gamma(m) \Gamma(n)}$

Hint:

Remember that the Beta function is symmetric, meaning $\beta(m, n) = \beta(n, m)$. This symmetry is clearly reflected in the symmetric numerator $\Gamma(m)\Gamma(n)$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Beta and Gamma functions are Euler integrals that are fundamental in advanced mathematical analysis and statistics.

Step 2: Detailed Explanation:
Let us review the standard mathematical relationship between these two functions:
For any positive real numbers $m > 0$ and $n > 0$, the Beta function $\beta(m, n)$ is defined as:
\[ \beta(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx \]
The Gamma function $\Gamma(z)$ is defined as:
\[ \Gamma(z) = \int_0^{\infty} e^{-t} t^{z-1} \, dt \]
The established mathematical theorem connects these two functions through the identity:
\[ \beta(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} \]
This relationship is crucial for solving complex probability density integral problems.
Therefore, option (C) is the correct relation.

Step 3: Final Answer
The correct option is (C).