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

calculate .. lim_( xā†’ 0^( +) ) āˆ«_0 ^( (š›‘/2)) J_0 ( x.cos(āˆ… )).cos(āˆ…)dāˆ… = ? where : J_š›Ž ( x )= x^( v) .Ī£_(n=0) ^āˆž (((āˆ’1 )^( n) .x^( 2n) )/(2^( 2n+v) .ššŖ (n + š›Ž +1 )))

$$ \\ $$$$\:\:\:\boldsymbol{{calculate}}\:.. \\ $$$$\:\boldsymbol{{lim}}_{\:\boldsymbol{{x}}\rightarrow\:\mathrm{0}^{\:+} } \int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{{J}}_{\mathrm{0}} \:\left(\:\boldsymbol{{x}}.\boldsymbol{{cos}}\left(\boldsymbol{\emptyset}\:\right)\right).\boldsymbol{{cos}}\left(\boldsymbol{\emptyset}\right)\boldsymbol{{d}\emptyset}\:=\:? \\ $$$$\:\:\:\boldsymbol{{where}}\:: \\ $$$$\:\:\:\:\:\boldsymbol{{J}}_{\boldsymbol{\nu}} \:\left(\:\boldsymbol{{x}}\:\right)=\:\boldsymbol{{x}}^{\:\boldsymbol{{v}}} .\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(āˆ’\mathrm{1}\:\right)^{\:\boldsymbol{{n}}} .\boldsymbol{{x}}^{\:\mathrm{2}\boldsymbol{{n}}} }{\mathrm{2}^{\:\mathrm{2}\boldsymbol{{n}}+\boldsymbol{{v}}} .\boldsymbol{\Gamma}\:\left(\boldsymbol{{n}}\:+\:\boldsymbol{\nu}\:+\mathrm{1}\:\right)} \\ $$

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