General Properties and related problems

General Properties and related problems

1. What "General Properties" Collects — Beta & Gamma Together

This capstone topic gathers all Gamma/Beta properties + duplication, reflection, and problem templates that combine earlier topics, plus general reduction tricks for multiple integrals that land on Beta/Gamma.

1. Domain: B(m,n) needs m>0,n>0 ; Γ(n) needs n>0 (else pole)
2. Symmetry: B(m,n)=B(n,m)
3. Gamma recurrence: Γ(n+1)=nΓ(n), Γ(1)=1, Γ(n+1)=n! , Γ(1/2)=√π
4. Beta recurrence: B(m+1,n)= m/(m+n) B(m,n)
5. Beta–Gamma bridge: B(m,n)= Γ(m)Γ(n)/Γ(m+n)
6. Trigonometric: ∫_{0}^{π/2} sin^{2m-1}θ cos^{2n-1}θ dθ =1/2 B(m,n)
7. Infinite / scale forms: ∫_{0}^{∞} x^{m-1}/(1+x)^{m+n} dx =B(m,n)=Γ(m)Γ(n)/Γ(m+n)
8. Duplication (Legendre): Γ(m)Γ(m+1/2)=2^{1-2m}√π Γ(2m)
9. Reflection (Euler): Γ(m)Γ(1-m)= π / sin(π m), 0<m<1 ⇒ Γ(1/2)=√π
10. Stirling (large m): Γ(m+1)≈ √(2π m) m^{m} e^{-m}

5. Problem-Solving Checklist

1. Recognise form $∫_{0}^{1} x^{m-1}(1-x)^{n-1}$ ⇒ $B$.
2. $∫_{0}^{∞} x^{n-1} e^{-a x}$ ⇒ $Γ(n)/a^{n}$.
3. $∫_{0}^{π/2} sin^{p} cos^{q}$ ⇒ $1/2 B((p+1)/2,(q+1)/2)$.
4. $x²+y²$ disc ⇒ polar + Gamma/Beta.
5. Quote $Γ(1/2)=√π, Γ(n+1)=n!$ directly.

6. Exam Points

• List general properties $B(m,n), Γ(n)$ — 5-7 marks, very frequent.
• Prove $B(m,n)=Γ(m)Γ(n)/Γ(m+n)$, duplication, reflection — 7 marks, very frequent.
• Evaluate mixed integrals $∫_{0}^{1} x^{m}/√(1-x^{n})$, $∫_{0}^{∞} x^{p}/(1+x^{q})$ via $B/Γ$ — 7 marks, very frequent.
• Solve 3-4 related problems combining double/triple with $B/Γ$ — 10 marks, very frequent (often section B compulsory).