Question Number 137420 by mnjuly1970 last updated on 02/Apr/21
$$\:\:\:\:\:\:\:\:\:\:\:\:\:......{mathematical}\:...\:...\:...\:{analysis}\left({II}\right)..... \\ $$$$\:\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\:\mathbb{R}} \left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}^{\mathrm{2}} \right)^{{n}} }{\left({n}!\right)^{\mathrm{2}} }\right){dx}=\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.......................... \\ $$
Commented by Dwaipayan Shikari last updated on 03/Apr/21
$$\int_{\mathbb{R}} \underset{{n}\geqslant\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}^{\mathrm{2}} \right)^{{n}} }{\left({n}!\right)^{\mathrm{2}{s}} }{dx}=\pi^{\mathrm{1}−{s}} \\ $$$$\int_{\mathbb{R}} \underset{{n}\geqslant\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}^{\mathrm{2}} \right)^{{n}} }{\left({n}!\right)^{\mathrm{2}} }=\mathrm{1} \\ $$