Using the relationship between beta and gamma functions, simplify B(m,n) B(m+n,P).
Understand the Problem
The question is asking to simplify an expression involving beta and gamma functions, particularly focusing on the relationship between them. It specifies the functions B(m,n) and B(m+n,P) and invites simplification using their properties.
Answer
$$ B(m, n) B(m+n, P) = \frac{\Gamma(m) \Gamma(n) \Gamma(P)}{\Gamma(m+n+P)} $$
Answer for screen readers
$$ B(m, n) B(m+n, P) = \frac{\Gamma(m) \Gamma(n) \Gamma(P)}{\Gamma(m+n+P)} $$
Steps to Solve
- Definition of Beta Function
The Beta function is defined using the Gamma function as follows: $$ B(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} $$
- Using the Definition for Both Functions
Now we'll express both $B(m, n)$ and $B(m+n, P)$ in terms of the Gamma function: $$ B(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} $$ $$ B(m+n, P) = \frac{\Gamma(m+n) \Gamma(P)}{\Gamma(m+n+P)} $$
- Combine the Two Beta Functions Together
Now we multiply these two expressions: $$ B(m, n) B(m+n, P) = \left(\frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}\right) \left(\frac{\Gamma(m+n) \Gamma(P)}{\Gamma(m+n+P)}\right) $$
- Simplifying the Expression
When we multiply the two, the $\Gamma(m+n)$ in the numerator and denominator will cancel out: $$ B(m, n) B(m+n, P) = \frac{\Gamma(m) \Gamma(n) \Gamma(P)}{\Gamma(m+n+P)} $$
$$ B(m, n) B(m+n, P) = \frac{\Gamma(m) \Gamma(n) \Gamma(P)}{\Gamma(m+n+P)} $$
More Information
The Beta function is significant in various areas of mathematics and statistics, both for its properties and its relationships with other functions like the Gamma function. This simplification showcases how these functions interact and can provide insights into integration and probability distributions.
Tips
- Failing to use the properties of the Gamma function when simplifying the product of Beta functions.
- Forgetting to cancel out terms correctly when multiplying fractions.
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