Gamma functions

Gamma functions

1. Gamma Function — Extended View (this topic overlaps "Gamma" but syllabus splits emphasis)

$Γ(n)=∫_{0}^{∞} x^{n-1} e^{-x} dx$ for $n>0$, with recurrence $Γ(n+1)=nΓ(n)$ and $Γ(n+1)=n!$ for integers, already detailed. This entry collects the formulas that use Gamma functions to evaluate integrals — the "Gamma-function technique".

2. Standard Gamma Integrals (memorise forms)

∫_{0}^{∞} x^{n-1} e^{-a x} dx = Γ(n)/a^{n} ,  a>0, n>0              (scale)
∫_{0}^{∞} x^{m} e^{-x²} dx = 1/2 Γ((m+1)/2)                          (Gaussian-type)
∫_{0}^{∞} e^{-x²} dx = √π/2 = Γ(1/2)/2
∫_{0}^{1} (-log x)^{n-1} dx = Γ(n)                                  (log substitution)
∫_{0}^{1} x^{m-1}(1-x)^{n-1} dx = B(m,n)=Γ(m)Γ(n)/Γ(m+n)            (Beta-Gamma link)

Derivation for $∫ x^{m} e^{-x²}dx$: put $t=x²$ ⇒ $x=√t, dx= dt/(2√t)$ ⇒ $∫ t^{m/2} e^{-t} dt/(2√t)=1/2 Γ((m+1)/2)$.

3. Worked Examples via Gamma Functions

(a) Evaluate ∫_{0}^{∞} x^5 e^{-3x} dx = Γ6 /3^6 =120/729=40/243 ✓

(b) ∫_{0}^{∞} x² e^{-x²} dx =1/2 Γ(3/2)=√π/4 ✓

(c) ∫_{0}^{∞} x^{1/2} e^{-2x} dx = Γ(3/2)/2^{3/2}= √π/(2·√2·2)? Compute: Γ3/2=√π/2 ⇒ / (2√2) ⇒ √π/(4√2) ✓

(d) Volume of n-ball? Uses Gamma: V_n = π^{n/2} a^{n}/Γ(n/2+1) ; for n=3 gives 4πa³/3 ✓

(e) Need to evaluate B(3.5,2.5)?? Use Gamma: =Γ3.5Γ2.5/Γ6 ; Γ3.5=15√π/8, Γ2.5=3√π/4 ⇒ product =45π/32 /120=3π/256? Check.

(f) ∫_{0}^{1} (x²(1-x²))^{3/2} dx? Put x²=t ⇒ uses Beta then Gamma.

4. Handling $0<n<1$ and Poles

$Γ(n)$ has simple poles at $0,-1,-2,...$ with $Γ(ε)≈1/ε -γ$ ($γ$ Euler). For $0<n<1$, use recurrence backwards: $Γ(n)=Γ(n+1)/n$, e.g., $Γ(1/3)= 3Γ(4/3)$.

5. Exam Points

• Evaluate $∫_{0}^{∞} x^{m} e^{-a x}dx, ∫_{0}^{∞} e^{-x²}dx$ via $Γ$ — 5 marks, very frequent.
• Show $∫_{0}^{∞} x^{m} e^{-x²}dx =1/2 Γ((m+1)/2)$ — 5 marks, frequent.
• Use $Γ$ to evaluate Beta integrals and trigonometric $∫_{0}^{π/2} sin^{m} cos^{n}$ — 7 marks, very frequent.
• State $Γ(1/2)=√π, Γ(n+1)=n!$ — 3 marks.