Beta function: B(a,b) = ∫₀¹ tᵃ⁻¹(1−t)ᵇ⁻¹dt = Γ(a)Γ(b)/Γ(a+b). Symmetric: B(a,b) = B(b,a). For integers: B(m,n) = (m−1)!(n−1)!/(m+n−1)! = 1/[n·C(m+n−1,m−1)]. Beta distribution: f(x) = xᵃ⁻¹(1−x)ᵇ⁻¹/B(a,b).
How to Use
Enter parameters:
- a, b: Positive real numbers
- B(a,b): Beta function value
- Related: Gamma function values
Properties
- B(a,b) = B(b,a)
- B(a,b) = 2∫₀^(π/2) sin²ᵃ⁻¹θ cos²ᵇ⁻¹θ dθ
- B(a,1) = 1/a
- B(1/2,1/2) = π
Applications
- Beta distribution in Bayesian statistics
- Binomial coefficient generalization
- Wallis integrals
- String theory amplitudes
Step-by-Step Instructions
- 1Enter a and b.
- 2Get B(a,b).
- 3View Γ(a), Γ(b).
- 4Check symmetry.
- 5See properties.