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

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