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Question Number 100047 by Ar Brandon last updated on 24/Jun/20

$${\int }_0^1{\mathrm{e}}^{-{\mathrm{x}}^2}\mathrm{dx}$$

Answered by smridha last updated on 24/Jun/20

$$=\frac{\sqrt{\mathit{\pi }}}{2}erf\left(1\right)$$

Commented by Ar Brandon last updated on 24/Jun/20

Thanks, but how do we arrive there ? Or is it just a theory ?

Commented by smridha last updated on 24/Jun/20

$$\mathit{y}\mathit{e}\mathit{a}\mathit{h}\mathit{y}\mathit{o}\mathit{u}\mathit{c}\mathit{a}\mathit{n}\mathit{s}\mathit{a}\mathit{y}\mathit{t}\mathit{h}\mathit{a}\mathit{t}$$$$\mathit{t}\mathit{h}\mathit{e}\mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{i}\mathit{s}..$$$$\mathit{e}\mathit{r}\mathit{f}\left(\mathit{x}\right)=\frac{2}{\sqrt{\mathit{\pi }}}{\int }_0^x{\mathit{e}}^{-{\mathit{t}}^2}\mathit{d}\mathit{t}$$$$\mathit{w}\mathit{e}\mathit{l}\mathit{l}\mathit{I}\mathit{h}\mathit{a}\mathit{v}\mathit{e}\mathit{d}\mathit{e}\mathit{f}\mathit{f}\mathit{e}\mathit{r}\mathit{e}\mathit{n}\mathit{t}\mathit{w}\mathit{a}\mathit{y}\mathit{w}\mathit{a}\mathit{i}\mathit{t}$$$$\mathit{I}\mathit{p}\mathit{o}\mathit{s}\mathit{t}\mathit{i}\mathit{t}...$$

Answered by smridha last updated on 24/Jun/20

$$={\int }_0^1\underset{\mathit{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{\mathit{x}}^{2\mathit{n}}}{\mathit{n}!}\mathit{d}\mathit{x}$$$$=1.\underset{\mathit{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{\mathit{n}}}{\mathit{n}!(2\mathit{n}+1)}$$$$=\underset{\mathit{n}=0}{\sum ^{\mathrm{\infty }}}\frac{\frac{1}{2}}{(\mathit{n}+\frac{1}{2})}.\frac{{(-1)}^{\mathit{n}}}{\mathit{n}!}$$$$=\underset{\mathit{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{\left(\frac{1}{2}\right)}_{\mathit{n}}}{{\left(\frac{3}{2}\right)}_{\mathit{n}}}.\frac{{(-1)}^{\mathit{n}}}{\mathit{n}!}$$$$=\mathit{M}(\frac{1}{2},\frac{3}{2},-1)\mathit{o}\mathit{r}{}_1{\mathit{F}}_1(\frac{1}{2};\frac{3}{2};-1)$$$$\mathit{t}\mathit{h}\mathit{i}\mathit{s}\mathit{i}\mathit{s}\mathit{c}\mathit{a}\mathit{l}\mathit{l}\mathit{e}\mathit{d}\mathit{c}\mathit{o}\mathit{n}\mathit{f}\mathit{l}\mathit{u}\mathit{e}\mathit{n}\mathit{t}\mathit{h}\mathit{y}\mathit{p}\mathit{e}\mathit{r}\mathit{g}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{i}\mathit{c}$$$${\mathit{f}}^{\mathit{n}}.$$$$\mathit{n}\mathit{o}\mathit{t}\mathit{e}:\left(\mathit{a}\right)=\frac{{\left(\mathit{a}\right)}_{\mathit{n}}}{{(\mathit{a}+1)}_{\mathit{n}}}(\mathit{a}+\mathit{n})\left[\mathit{p}\mathit{o}\mathit{c}\mathit{h}\mathit{h}\mathit{a}\mathit{m}\mathit{m}\mathit{e}\mathit{r}\mathit{s}\mathit{y}\mathit{m}\mathit{b}\mathit{o}\mathit{l}\right]$$

Commented by Ar Brandon last updated on 24/Jun/20

Thanks so very much��

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