EDB — 1BC

Exercises

1. [1BC]We define the Beta function as

   \[ B(x,y) = ∫_ 0^ 1 t^{x-1}(1-t)^{y-1}~ dt~ . \]

   1. Show that the integral exists (finite) if and only if \(x,y{\gt}0\).

   2. Note that \(B(x,y)=B(y,x)\)

   3. Relate \(B(n,m)\) to \(B(n-1,m+1)\). Then calculate the value of \(B(n,m)\) for \(n,m\) natural positives.

   4. Use the result to calculate

      \[ ∫_ 0^{𝜋/2} \sin (t)^ 9\cos (t)^ 7~ dt~ ~ . \]
   Solution 1

   [1BD]↺↻