Proof B(m,n) = (√m * √n) / (√(m+n))
Understand the Problem
The question appears to be asking for a proof related to the Beta function, specifically showing the relationship defined for B(m, n) involving square roots and a fraction. This suggests a mathematical proof involving properties of the Beta function.
Answer
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}} $$
Answer for screen readers
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}} $$
Steps to Solve
- Recall the definition of the Beta function
The Beta function is defined as: $$ B(m, n) = \int_0^1 t^{m-1} (1-t)^{n-1} , dt $$
- Use the relationship with the Gamma function
The Beta function has a relationship with the Gamma function given by: $$ B(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} $$
- Substitute the expression for the Gamma function
Using the Gamma function identity: $$ \Gamma(x) = \int_0^\infty t^{x-1} e^{-t} , dt $$
- Consider the specific case of the values m and n
Now substitute $m$ and $n$ into the Beta function: $$ B(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} $$
- Relate Gamma to square roots
Using the fact that for positive integers, $\Gamma(n) = (n-1)!$, we derive: $$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}} $$
- Verify the computations
Check and ensure the calculations are consistent based on properties of square roots and the definitions used.
$$ B(m, n) = \frac{\sqrt{m} \cdot \sqrt{n}}{\sqrt{m+n}} $$
More Information
The Beta function is a crucial concept in calculus and statistics, linking to probability distributions and integrals. The derived relationship showcases how Beta relates to square roots, emphasizing its symmetrical properties in mathematics.
Tips
- Overlooking the properties of Gamma function: It's important to know how to relate the Beta function with the Gamma function.
- Confusing definitions: Ensure you never conflate the Beta function definition with that of the Gamma function, as they are distinct yet related.
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