Relation between Beta

Relation between Beta

1. Beta–Gamma Relation (the relation examiners mean)

For $m>0,n>0$:

is the bridge between the first and second Eulerian integrals. This single formula lets you evaluate any Beta via factorials / $√π$.

Proof sketch (for exam, outline): write $Γ(m)Γ(n)= ∫_{0}^{∞}∫_{0}^{∞} x^{m-1}y^{n-1} e^{-(x+y)}dx dy$, set $x= r² cos²θ$ type transformation or $x= uv, y= u(1-v)$ ⇒ Jacobian gives $∫_{0}^{∞} u^{m+n-1} e^{-u}du · ∫_{0}^{1} v^{m-1}(1-v)^{n-1}dv = Γ(m+n) B(m,n)$.

B(m,n)=B(n,m) follows from Gamma symmetry.
B(m+1,n)= m/(m+n) B(m,n) since Γ(m+1)=mΓ(m)
B(1/2,1/2)= Γ(1/2)²/Γ1 = π ; matches direct integral ∫0^1 dx/√(x(1-x))=π

4. Other Relations Involving Beta

• Trigonometric:

• Infinite:

• With symmetry $B(m,n) = 2∫_{0}^{π/2}...$ as in topic 9.

5. Exam Points

• State and prove $B(m,n)=Γ(m)Γ(n)/Γ(m+n)$ (sketch) — 5-7 marks, very frequent.
• Use it to evaluate $∫_{0}^{π/2} sin^{p} cos^{q}$ or $∫_{0}^{1} x^{m}(1-x)^{n}dx$ — 5 marks, very frequent.
• Show $B(1/2,1/2)=π$ — 3 marks, frequent.
• Derive $B(m+1,n)=m/(m+n)B(m,n)$ from relation — 3 marks.