A question relating trigonometric integrals with gamma function

Question

Why is it that $$\int _0 ^{\frac {\pi}{2}} \sin^m (x)\cos^n(x) =\frac {\gamma\left(\frac {m+1}{2}\right).\gamma \left(\frac {n+1}{2}\right)}{\gamma \left(\frac {m+n+2}{2}\right)} $$. How is it proved. Further my professor added that its true only for integers $m,n $. Why so? Also he said that it was only for this specific form and not for any other like where you have argument of sine,cosine as $2x $. I know we can substitute $2x=u $ and then continue but why cant we carry on with the same $2x $ as we do it for $x $. Is it associated with period of $\sin,\cos$. Thanks!

Answer 1

Your prof is not correct. It is defined for some real values. Also your notation is used for Incomplete gamma function.

Start by

$$\int^1_0 t^{x-1} (1-t)^{y-1}\,dt = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$

Let $t = \sin^2 \theta$

$$2\int^{\pi/2}_0 \sin^{2x-1}(\theta) \cos^{2y-1}(\theta)\,d\theta = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$

This is defined for $\Re \, x , y > 0$. You can let $2x-1 = n,2y-1 =m$.

For instace we have $x=3/4,y=1/2$

$$\int^{\pi/2}_0 \sqrt{\sin \theta}\, d\theta = \sqrt{\frac{2}{\pi}}\Gamma^2\left(\frac{3}{4} \right)$$

We can also prove the somewhat similar identity

$$\int^{\pi/2}_{0}\cos(x \theta)\cos^y(\theta)\,d\theta=\frac{\pi \Gamma(y+1)}{2^{y+1}\Gamma\left(\frac{x+y+2}{2}\right)\Gamma\left(\frac{2-x+y}{2}\right)}$$

using contour integration.