Define Beta and Gamma functions

Gamma function

For n > 0, the Gamma function is defined by the improper integral

Γ(n) = ∫₀^∞ e^(−x) x^(n−1) dx.

Key properties:

• Γ(n + 1) = n Γ(n), so Γ(n + 1) = n! for a positive integer n.
• Γ(1) = 1, Γ(1/2) = √π.
• Γ(n)Γ(1 − n) = π / sin(nπ), 0 < n < 1.

Beta function

For m, n > 0, the Beta function (Eulerian integral of the first kind) is

B(m, n) = ∫₀¹ x^(m−1) (1 − x)^(n−1) dx.

Key properties:

• Symmetry: B(m, n) = B(n, m).
• Trigonometric form: B(m, n) = 2∫₀^(π/2) sin^(2m−1)θ cos^(2n−1)θ dθ.
• Relation with Gamma: B(m, n) = Γ(m) Γ(n) / Γ(m + n).

Example: B(2, 3) = Γ(2)Γ(3)/Γ(5) = (1·2)/24 = 1/12.