Enter two positive real numbers — x (alpha) and y (beta) — and the Beta Function Calculator computes B(x, y) using the integral definition via the Gamma function relationship. Choose your desired decimal precision and get the exact value along with key properties like symmetry and the Gamma function breakdown.
B(x, y)
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Γ(x)
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Γ(y)
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Γ(x + y)
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B(y, x) — Symmetry Check
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The Beta function B(x, y) is a special mathematical function defined by the integral B(x, y) = ∫₀¹ t^(x−1) · (1−t)^(y−1) dt, where both x and y must be positive real numbers. It arises frequently in probability theory, statistics, and combinatorics.
The Beta function can be expressed in terms of the Gamma function as B(x, y) = Γ(x) · Γ(y) / Γ(x + y). This relationship is extremely useful for computation since the Gamma function has well-known numerical approximation methods such as the Lanczos approximation.
Both parameters x (alpha) and y (beta) must be strictly positive real numbers — that is, greater than zero. The function is not defined for zero or negative integer inputs, as the Gamma function has poles at non-positive integers.
The symmetry property B(x, y) = B(y, x) follows directly from the integral definition by substituting t → 1 − t. This means swapping the two parameters does not change the result, which is confirmed by the Gamma function formula since multiplication is commutative.
B(1/2, 1/2) equals π (pi), approximately 3.14159. This is a well-known special value that connects the Beta function to the constant π through the Gamma function identity Γ(1/2) = √π.
The Beta function is used in probability and statistics — most notably as the normalizing constant in the Beta distribution. It also appears in Bayesian inference, combinatorics, number theory, and physics, including the computation of volume integrals and quantum field theory calculations.
The Beta distribution is a continuous probability distribution defined on the interval [0, 1], parameterized by α and β. Its probability density function uses the Beta function B(α, β) as its normalizing constant, ensuring the distribution integrates to 1 over [0, 1].
The Beta function is computed numerically using the Lanczos approximation for the Gamma function. Higher precision settings (up to 12 decimal places) display more significant digits of the result, which is useful in scientific computing contexts where high accuracy is required.