Calculate the Beta function B(m, n) using the Gamma function relationship.
How It Works
How Beta Function Calculator Works
The Beta function is defined as B(m, n) = Γ(m)·Γ(n) / Γ(m+n), where Γ is the Gamma function — a continuous extension of the factorial (Γ(k) equals (k−1)! for whole numbers). The calculator evaluates Gamma using the Lanczos approximation, a well-established numerical method accurate to many decimal places.
Worked Example
See It In Action
B(1, 4) = Γ(1)·Γ(4) / Γ(5) = 1 × 6 / 24 = 0.25, using the fact that Γ(1)=0!=1, Γ(4)=3!=6, and Γ(5)=4!=24.
FAQ
Frequently Asked Questions
Where is the Beta function actually used?
It shows up in probability and statistics (the Beta probability distribution is built directly from it) as well as in calculus and combinatorics problems involving integrals of the form ∫x^(m-1)(1-x)^(n-1)dx.
Why use the Gamma function instead of factorials directly?
Factorials are only defined for non-negative integers, but the Gamma function extends the same idea to any positive real number (and beyond), which is what lets m and n be non-integer values like 2.5.
