Gamma

Gamma

1. Definition — Gamma Function (Second Eulerian Integral)

For $n>0$ (real):

is the Gamma function. It converges for $n>0$ (for $n≤0$ it needs analytic continuation; poles at $0,-1,-2,...$). For integers, it extends factorial.

2. Properties

• Recurrence / functional equation:

(integration by parts).

• Factorial connection:

since $Γ(1)=1$, so $Γ(1)=1, Γ(2)=1, Γ(3)=2, Γ(4)=6$ etc.

• Half-integers:

e.g., $Γ(3/2)=√π/2, Γ(5/2)=3√π/4$.

• Reflection (state):

— gives $Γ(1/2)=√π$.

• Special values:

.

3. Worked Examples

(a) Γ(5)=4! =24 ; Γ(6)=120 ; Γ(1/2)=√π ≈1.772 ✓

(b) I = ∫_{0}^{∞} x⁴ e^{-x} dx = Γ5 =24 ✓

(c) I = ∫_{0}^{∞} x² e^{-x²} dx : put t=x² ⇒ x=√t, dx= dt/(2√t) ⇒ I=1/2 ∫ t^{1/2} e^{-t} dt =1/2 Γ(3/2)= √π/4 ✓

(d) I = ∫_{0}^{∞} x^{3} e^{-2x} dx : let u=2x ⇒ =1/16 ∫ u³ e^{-u} du =Γ4/16=6/16=3/8 ✓

(e) Need to evaluate B via Gamma: B(1/2,1/2)=Γ(1/2)Γ(1/2)/Γ1=π ✓ matches earlier.

(f) ∫_{0}^{∞} e^{-x⁴} dx? Let t=x⁴ ⇒ dx= t^{-3/4} dt/4 ⇒ =Γ(1/4)/4 ✓

4. Relation to Factorials

Gamma interpolates factorial to non-integers: $n! = Γ(n+1)$ holds for $n$ real $>-1$. Stirling's approximation for large $n$ uses $Γ$.

5. Exam Points

• Define $Γ(n)=∫_{0}^{∞} x^{n-1} e^{-x}dx, n>0$ — 3 marks, very frequent.
• Prove $Γ(n+1)=nΓ(n)$ and $Γ(n+1)=n!$ — 5 marks, very frequent.
• Evaluate $Γ(1/2), Γ(3/2), Γ(5/2)$ — 3-5 marks, frequent.
• Use $Γ$ to evaluate $∫_{0}^{∞} x^{n} e^{-a x}dx = Γ(n+1)/a^{n+1}$ — 5 marks, very frequent.
• Show $B(m,n)=Γ(m)Γ(n)/Γ(m+n)$ — 5 marks (in relation topic).