Beta Function Calculator B(x, y) and Incomplete Beta

Euler's Beta function B(x, y) calculator: Gamma relation, logarithmic form, incomplete Beta and regularized I_x(a,b) as Beta distribution CDF for statistics.

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The Beta function

The Beta function, introduced by Leonhard Euler in 1730 alongside Gamma, is a two-parameter integral defined as:

B(x, y) = integral_0^1 t^(x-1) * (1-t)^(y-1) dt, for Re(x), Re(y) > 0

It is closely linked to Gamma via the fundamental identity:

B(x, y) = Gamma(x) * Gamma(y) / Gamma(x + y)

Appears in probability (Beta distribution), statistics (hypothesis tests), quantum physics (Veneziano formula), number theory and combinatorics (fractional binomial coefficients).

Symmetry and notable values

Beta is symmetric: B(x, y) = B(y, x). Notable values:

  • B(1, 1) = 1 (constant integrand 1 on [0, 1])
  • B(1/2, 1/2) = pi (substitution t = sin^2(u))
  • B(1, n) = 1/n for n positive integer
  • B(m, n) = (m-1)! * (n-1)! / (m+n-1)! for m, n positive integers
  • B(a, 1) = 1/a for any a > 0

Relation to binomial coefficients

Beta generalizes binomials to reals or complex numbers. For integers m, n:

C(m+n-1, m-1) = 1 / (m * B(m, n))

Generalized binomial C(a, k) = 1 / ((a+1) * B(k+1, a-k+1)) extends to fractions and complex, fundamental in statistics for Fisher distribution and Student's t-test.

Incomplete Beta function

Incomplete Beta generalizes to variable integration limit:

B(x; a, b) = integral_0^x t^(a-1) * (1-t)^(b-1) dt, for 0 <= x <= 1

Regularized version divides by complete value:

I_x(a, b) = B(x; a, b) / B(a, b)

I_x(a, b) is the CDF of Beta(a, b) distribution. Values I_0(a,b) = 0, I_1(a,b) = 1. Symmetry: I_x(a, b) = 1 - I_{1-x}(b, a).

Numerical computation: continued fraction

Calculator uses Lentz-Thompson continued fraction for I_x(a, b) with quadratic convergence. For x < (a+1)/(a+b+2) computes directly, else uses symmetry I_x(a,b) = 1 - I_{1-x}(b,a) to avoid numerical slowness. Combined with Lanczos for ln Gamma, yields 12-15 decimals in 50-100 iterations.

Five modes

  1. B(x, y) direct: via exp(ln Gamma(x) + ln Gamma(y) - ln Gamma(x+y)).
  2. ln B(x, y): for large parameters avoids overflow (B can be 10^-300 or 10^300).
  3. Incomplete Beta: B(x; a, b) with truncated integration.
  4. Regularized Beta: I_x(a, b) = CDF of Beta(a, b) distribution.
  5. Binomial coefficient: C(a, k) generalized via Beta.

Statistics applications

Beta(alpha, beta) distribution has density f(x) = x^(alpha-1)(1-x)^(beta-1) / B(alpha, beta) on [0, 1]. Used for:

  • Bayesian inference: conjugate priors for binomial proportions.
  • Reliability analysis: system success probability.
  • Dirichlet distribution: multivariate generalization of Beta.
  • Fisher F-test: based on variance ratio, CDF expressible via I.
  • Student's t-test: equivalent to I_x(df/2, 1/2) with proper transformation.
  • Random variable mapping between [0,1] and arbitrary intervals.

Veneziano formula and string theory

Gabriele Veneziano (1968) described scattering amplitude of two particles using Beta:

A(s, t) = B(-alpha(s), -alpha(t)) with alpha(x) = 1/2 + (1/2) * x

Veneziano's formula has notable properties (s-t duality) suggesting existence of 'strings' as fundamental particles. Started modern string theory. Beta also appears in N-point calculations in string theory via Koba-Nielsen formulas.

Physics applications

  • Particle physics: Veneziano formula (1968), bosonic string theory.
  • Statistical mechanics: Dirichlet distributions for composition mixtures.
  • Thermodynamics: partition integrals for constrained systems.
  • Numerical analysis: Gauss-Jacobi quadrature uses Beta-like weights.
  • Calculus of variations: functionals of Beta integral form.
  • Image processing: Beta distribution for modeling pixel intensity.

History of the Beta function

The Beta function was introduced by Leonhard Euler in the 18th century, alongside the more famous Gamma function, while studying the integrals that extend the factorial. For this reason the Beta is also known as the Euler integral of the first kind, while the Gamma is that of the second kind. Euler discovered its integral definition and its deep link with the Gamma function, expressed by the relation B(x, y) = Γ(x)·Γ(y)/Γ(x+y). The name and the notation B were later adopted by Jacques Binet and Adrien-Marie Legendre in the 19th century. From a tool born to generalise binomial coefficients and compute integrals, the Beta function has become one of the most used special functions in mathematics, with applications ranging from statistics to particle physics.

The integral representations of the Beta function

The Beta function admits several equivalent integral forms, each useful in different contexts. The classic definition is B(x, y) = ∫₀¹ t^(x−1)·(1−t)^(y−1) dt, an integral over the unit interval that converges for positive x, y. With the substitution t = sin²θ you obtain the trigonometric form B(x, y) = 2·∫₀^(π/2) sin^(2x−1)θ · cos^(2y−1)θ dθ, particularly useful for computing integrals of powers of sine and cosine. There is also a form over the whole positive half-line, B(x, y) = ∫₀^∞ u^(x−1)/(1+u)^(x+y) du. These representations connect the Beta function to the Wallis product, to Gaussian integrals and to numerous formulas of analysis.

Updated July 2026

Frequently Asked Questions

What is the Beta function?

The Beta function B(x, y) is a special function defined by the integral ∫₀¹ t^(x−1)(1−t)^(y−1) dt for positive x, y. It is closely linked to the Gamma function by the relation B(x, y) = Γ(x)·Γ(y)/Γ(x+y). Introduced by Euler, it generalises binomial coefficients and appears in statistics, probability and physics. It is also called the Euler integral of the first kind.

Why is the Beta function symmetric?

Because swapping its two arguments does not change the value: B(x, y) = B(y, x). This follows directly from the relation with the Gamma, B(x, y) = Γ(x)Γ(y)/Γ(x+y), in which the numerator is symmetric in x and y. In the integral definition, the symmetry corresponds to the change of variable t → 1−t, which swaps the roles of the two exponents while leaving the integral unchanged.

How does it relate to binomial coefficients?

The Beta function generalises binomial coefficients to real numbers. The relation C(a, k) = 1/((a+1)·B(k+1, a−k+1)) holds, allowing the binomial coefficient to be defined even for non-integer arguments. For integer values the classic coefficients of Pascal's triangle are recovered. This connection makes the Beta a natural tool in combinatorics and probability calculus.

What is the incomplete Beta function?

It is a generalisation in which the integral stops at a value t instead of reaching 1: B(t; x, y) = ∫₀ᵗ u^(x−1)(1−u)^(y−1) du. Dividing it by the complete Beta gives the regularised incomplete Beta function I_t(x, y), which equals 0 at t=0 and 1 at t=1. The latter is exactly the cumulative distribution function of the Beta distribution, central in statistics.

How is the incomplete Beta computed numerically?

A continued fraction expansion is used, a classic method of numerical analysis (Numerical Recipes algorithm). To ensure rapid convergence the symmetry relation I_t(a, b) = 1 − I_{1−t}(b, a) is exploited, choosing the form that converges faster depending on whether t is less than or greater than (a+1)/(a+b+2). The method reaches high precision in a few iterations.

What role does the Beta play in the statistical distribution?

The Beta distribution, defined on the interval [0, 1], uses the Beta function as a normalisation constant: its density is proportional to t^(a−1)(1−t)^(b−1), divided by B(a, b). It is fundamental in Bayesian inference, where it models probabilities themselves, and it is the conjugate distribution of the binomial. The regularised incomplete Beta provides its cumulative distribution function.

What is the Veneziano formula?

It is a formula proposed by the physicist Gabriele Veneziano in 1968 to describe the scattering amplitude of particles in the strong interaction, expressed precisely through Euler Beta functions. Its remarkable symmetry led to the birth of string theory: it was discovered that the Veneziano amplitude naturally describes the vibration of relativistic strings. It is a famous example of how a classic mathematical function opened a revolution in physics.

Who introduced the Beta function?

It was Leonhard Euler in the 18th century, alongside the Gamma function, while studying the integrals that extend the factorial. The Beta is therefore called the Euler integral of the first kind. The notation B and the modern name took hold in the 19th century with Binet and Legendre. Since then the Beta function has been a standard tool of analysis, with applications in statistics, probability, combinatorics and theoretical physics.

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