$\displaystyle\overbrace{\idotsint}^{k}\limits_{\mathbb{F}}\prod_{n=1}^k\bigg(a_n^{b_n-1}\:da_n\bigg)=\frac{\prod_{n=1}^k\:\Gamma\Big(\frac{b_n}{2}\Big)}{2^k\:\Gamma\bigg(\Big[\sum_{n=1}^k\frac{b_n}{2}\Big]+1\bigg)}$

Question

$$\large\overbrace{\idotsint}^{k}\limits_{\mathbb{F}}\prod_{n=1}^k\bigg(a_n^{b_n-1}\:da_n\bigg)=\frac{\prod_{n=1}^k\:\Gamma\Big(\frac{b_n}{2}\Big)}{2^k\:\Gamma\bigg(\Big[\sum_{n=1}^k\frac{b_n}{2}\Big]+1\bigg)}\\\mathbb{F}:\sum_{n=1}^k\:a_n^2\leq1\:\:\:,\:\:\:\forall\:n:a_n\ge0\\k\in\mathbb{Z}^+\:\:\:,\:\:\:k>1$$

Eşitliğinin doğru olup olmadığını gösteriniz.

Comments

$k$ değerinin $2$ ve $3$ olduğu örnekler :

$\large\:k=2$

$\large\:k=3$