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Question Number 175080 by mnjuly1970 last updated on 18/Aug/22

$$$$$$\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}..$$$${\mathit{l}\mathit{i}\mathit{m}}_{\mathit{x}\to 0^{+}}{\int }_0^{\frac{\mathit{\pi }}{2}}{\mathit{J}}_0(\mathit{x}.\mathit{c}\mathit{o}\mathit{s}\left(\mathbf{\varnothing }\right)).\mathit{c}\mathit{o}\mathit{s}\left(\mathbf{\varnothing }\right)\mathit{d}\mathbf{\varnothing }=?$$$$\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}:$$$${\mathit{J}}_{\mathit{\nu }}\left(\mathit{x}\right)={\mathit{x}}^{\mathit{v}}.\underset{\mathit{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{\mathit{n}}.{\mathit{x}}^{2\mathit{n}}}{2^{2\mathit{n}+\mathit{v}}.\mathbf{\Gamma }(\mathit{n}+\mathit{\nu }+1)}$$

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