Beta

Beta

1. Definition — Beta Function (First Eulerian Integral)

For $m>0, n>0$ (real):

is the Beta function. It is also called the first Eulerian integral. $B(m,n)$ converges for $m>0, n>0$ (otherwise singularity at $0$ or $1$ diverges).

Trigonometric form (often used): putting $x = sin²θ$ ($dx =2 sinθ cosθ dθ$) gives:

so $∫_{0}^{π/2} sin^{p}θ cos^{q}θ dθ = 1/2 B((p+1)/2,(q+1)/2)$.

2. Properties

• Symmetry:

• Relation to Gamma (next topic, but state here):

• Recurrence:

• Special values:

3. Worked Examples

(a) Evaluate B(3,4)= ∫_{0}^{1} x²(1-x)³ dx = Γ3Γ4/Γ7 =2!·3!/6! =12/720=1/60 ✓

(b) I = ∫_{0}^{π/2} sin^5 θ cos^3 θ dθ = 1/2 B(3,2)=1/2·Γ3Γ2/Γ5=1/2·2·1/24=1/24 ✓

(c) I = ∫_{0}^{1} x^3/(1 -? Actually ∫0^1 x^m(1-x)^n type: ∫0^1 x^4(1-x)^2 dx = B(5,3)=4!2!/7! =48/5040=1/105 ✓

(d) Need to watch: ∫_{0}^{a} x^{m-1}(a -x)^{n-1} dx = a^{m+n-1} B(m,n) via x=a t ✓

(e) ∫_{0}^{∞} x^{m-1}/(1+x)^{m+n} dx = B(m,n) (another form) ✓

4. Use

Converts trigonometric / algebraic integrals to Gamma ratios, which are easier to evaluate for integer/half-integer $m,n$ (using $Γ(n)=(n-1)!, Γ(n+1/2)$ values). Essential for double/triple integrals over spheres and for proving Stirling-type results.

5. Exam Points

• Define $B(m,n)$ with condition $m>0,n>0$ — 3 marks, very frequent.
• Trigonometric form — 3-5 marks, frequent.
• Evaluate $B(2,3), ∫_{0}^{π/2} sin^m cos^n$ via $B$ — 5 marks, very frequent.
• Show $B(m,n)=B(n,m)$ and $B(m+1,n)=m/(m+n)B(m,n)$ — 5 marks, frequent.