Beta Function Calculator

About Beta Function Calculator

The beta function B(a,b) = Γ(a)Γ(b)/Γ(a+b). For positive integers, B(m,n) = (m-1)!(n-1)!/(m+n-1)!.

How to use this calculator

1. Enter a positive value for \(a\).
2. Enter a positive value for \(b\).
3. Use the calculator to evaluate \(B(a,b)\) from the gamma function formula.
4. Check the result against a known value or a simplified case if needed.

The formula explained

The beta function is defined by \(B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\). This formula computes the beta value for positive inputs \(a\) and \(b\).

• a = the first positive input to the beta function
• b = the second positive input to the beta function
• \(\Gamma(x)\) = the gamma function, which extends factorial-like values to non-integers

Step by step method

1. Write the formula \(B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\).
2. Find \(\Gamma(a)\), \(\Gamma(b)\), and \(\Gamma(a+b)\).
3. Substitute those values into the fraction and simplify.

Worked example

Problem. Find \(B(2,3)\).

1. Use the formula, \(B(2,3)=\frac{\Gamma(2)\Gamma(3)}{\Gamma(5)}\).
2. Evaluate the gamma values, \(\Gamma(2)=1\), \(\Gamma(3)=2\), and \(\Gamma(5)=24\).
3. Substitute and simplify, \(B(2,3)=\frac{1\cdot 2}{24}=\frac{1}{12}\).

Tips and common mistakes

• Make sure both inputs are positive, because the standard beta function is defined for positive values here.
• Remember that \(\Gamma(n)=(n-1)!\) for positive integers, so some examples can be simplified quickly.

Frequently asked questions