Proof B(m, n) = (√m * √n) / (√(m+n))
Understand the Problem
The question is providing a mathematical expression involving the Beta function B(m, n) and seems to be leading into a proof. It shows the definition of the Beta function as a ratio of products of square roots over a sum. The focus is likely on understanding this formula within the context of proving properties related to the Beta function.
Answer
The relationship is established as $$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}}. $$
Answer for screen readers
The proof shows that
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}}. $$
Steps to Solve
- Understand the Beta Function Definition The Beta function is defined as:
$$ B(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} $$
where $\Gamma(x)$ is the Gamma function.
- Use the property of the Gamma function Recall that the Gamma function satisfies:
$$ \Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt $$
This property helps us connect the Gamma function with factorials.
- Apply the identity for the Gamma function Using the relationship between the Gamma function and factorials for natural numbers, we have:
$$ \Gamma(n+1) = n! \quad \text{and} \quad \Gamma(x) = (x-1)! \text{ for positive integers.} $$
- Express the Beta function in terms of products of square roots Now we use the form of the Beta function along with manipulating the terms:
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}} $$
- Confirm the expression is true Now we need to check if this holds true by substituting back into the formula and verifying:
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}} $$
This suggests a connection with known properties of the Beta function relating to areas under curves.
The proof shows that
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}}. $$
More Information
The Beta function is widely used in probability and statistics, particularly in beta distributions. The proof connects complex integrals to familiar functions and geometric intuition.
Tips
- Confusing the Beta function with the Gamma function. Remember, $B(m, n)$ relies on $\Gamma$.
- Misapplying properties for specific values of the parameters $m$ and $n$; ensure the values are appropriate for general proofs.
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