Beta Function Calculator
Calculate the beta function B(x,y) with step-by-step solutions using gamma function relationships
B(x, y) = Γ(x)Γ(y) / Γ(x+y)
Integral form: B(x, y) = ∫₀¹ t^(x-1)(1-t)^(y-1) dt
Both x and y must be positive real numbers (> 0)
Quick Examples
Invalid Input
Both x and y must be positive real numbers (greater than 0).
B(, )
= Γ()Γ() / Γ()
Γ()
Γ()
Γ()
Reciprocal 1/B(x,y)
Log Beta ln(B(x,y))
Step-by-Step Solution
Identify the input values
x = , y =
Calculate Gamma function values
Γ() =
Γ() =
Γ() =
Apply the Beta function formula
B(x, y) = Γ(x)Γ(y) / Γ(x+y)
Result
B(, ) = × / =
Beta Function Properties
Symmetry
B(x, y) = B(y, x)
Recurrence
B(x+1, y) = B(x,y) × x/(x+y)
Special Value
B(1/2, 1/2) = π
For Integers
B(m,n) = (m-1)!(n-1)!/(m+n-1)!
Special Values Reference
| x | y | B(x, y) | Notes |
|---|---|---|---|
| 1 | 1 | 1 | Base case |
| 1/2 | 1/2 | π ≈ 3.14159 | Famous special value |
| 2 | 3 | 1/12 ≈ 0.0833 | = 1!×2!/4! |
| 3 | 4 | 1/60 ≈ 0.0167 | = 2!×3!/6! |
| 1 | n | 1/n | Simple reciprocal |
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About Beta Function Calculator
What is the Beta Function?
The Beta function B(x, y), also called Euler's integral of the first kind, is a fundamental mathematical function that takes two positive real numbers as arguments and produces a real number.
Mathematical Definition
The Beta function is defined by the integral:
B(x, y) = ∫₀¹ t^(x-1)(1-t)^(y-1) dt
Relation to Gamma Function
The Beta function can be expressed in terms of the Gamma function:
B(x, y) = Γ(x)Γ(y) / Γ(x+y)
For positive integers:
B(p, q) = (p-1)!(q-1)! / (p+q-1)!
Key Properties
Symmetry Property
B(x, y) = B(y, x)
The order of arguments doesn't affect the result.
Recurrence Relations
- B(x+1, y) = B(x, y) × x/(x+y)
- B(x, y+1) = B(x, y) × y/(x+y)
- B(x+1, y) + B(x, y+1) = B(x, y)
Special Values
- B(1, 1) = 1
- B(1/2, 1/2) = π
- B(1, n) = 1/n
- B(m, n) = 1/(m × C(m+n-1, n-1)) where C is the binomial coefficient
Trigonometric Form
B(x, y) = 2 ∫₀^(π/2) (sin θ)^(2x-1)(cos θ)^(2y-1) dθ
Applications
- Probability and Statistics - Central to the Beta distribution
- Bayesian Inference - Prior and posterior distributions
- Combinatorics - Related to binomial coefficients
- Physics - String theory scattering amplitudes
- Machine Learning - Beta-binomial models, Dirichlet distributions
Relation to Binomial Coefficients
B(m, n) = 1 / (n × C(m+n-1, m-1))
where C(n, k) is the binomial coefficient "n choose k".
Common Mistakes
- Confusing with Beta distribution - B(x,y) is the function, Beta(α,β) is the probability distribution that uses it
- Forgetting symmetry - B(2,3) = B(3,2)
- Invalid inputs - Both x and y must be positive real numbers