Question Number 82073 by TawaTawa last updated on 18/Feb/20 | ||
$$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\mathrm{j}_{\mathrm{3}/\mathrm{2}} \left(\mathrm{x}\right)\:\:=\:\:\frac{\sqrt{\mathrm{2}}}{\pi\mathrm{x}}\:\left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\:−\:\mathrm{cos}\:\mathrm{x}\right) \\ $$ | ||
Commented by jagoll last updated on 18/Feb/20 | ||
$${what}\:{is}\:{j}_{\frac{\mathrm{3}}{\mathrm{2}}} ? \\ $$ | ||
Commented by TawaTawa last updated on 18/Feb/20 | ||
Commented by TawaTawa last updated on 18/Feb/20 | ||
$$\mathrm{Am}\:\mathrm{lost}\:\mathrm{here}. \\ $$ | ||
Answered by mind is power last updated on 18/Feb/20 | ||
$${J}_{{a}} \left({x}\right)=\left(\frac{{x}}{\mathrm{2}}\right)^{{a}} \underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{k}} {x}^{\mathrm{2}{k}} }{\mathrm{2}^{\mathrm{2}{k}} {k}!.\Gamma\left({k}+\mathrm{1}+{a}\right)} \\ $$$${j}_{\frac{\mathrm{3}}{\mathrm{2}}} \left({x}\right)=\left(\frac{{x}}{\mathrm{2}}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{k}} {x}^{\mathrm{2}{k}} }{\mathrm{2}^{\mathrm{2}{k}} {k}!\Gamma\left({k}+\mathrm{1}+\frac{\mathrm{3}}{\mathrm{2}}\right)} \\ $$$$\Gamma\left({k}+\mathrm{1}+\frac{\mathrm{3}}{\mathrm{2}}\right)=\left({k}+\frac{\mathrm{3}}{\mathrm{2}}\right)\left({k}+\frac{\mathrm{1}}{\mathrm{2}}\right)........\frac{\mathrm{1}}{\mathrm{2}}.\sqrt{\pi} \\ $$$$=\frac{\left(\mathrm{2}{k}+\mathrm{3}\right).....\left(\mathrm{2}{k}+\mathrm{1}\right).....\sqrt{\pi}}{\mathrm{2}^{{k}+\mathrm{2}} } \\ $$$${j}_{\frac{\mathrm{3}}{\mathrm{2}}} \left({x}\right)=\frac{\mathrm{1}}{{x}\sqrt{{x}}}\frac{\sqrt{\mathrm{2}}}{\sqrt{\pi}}\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{k}} {x}^{\mathrm{2}{k}+\mathrm{3}} }{\left(\mathrm{2}{k}+\mathrm{3}\right).\left(\mathrm{2}{k}+\mathrm{1}\right)!} \\ $$$${f}\left({x}\right)=\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{k}} {x}^{\mathrm{2}{k}+\mathrm{3}} }{\left(\mathrm{2}{k}+\mathrm{3}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)!} \\ $$$${f}'\left({x}\right)=\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{k}} {x}^{\mathrm{2}{k}+\mathrm{2}} }{\left(\mathrm{2}{k}+\mathrm{1}\right)!}={xsin}\left({x}\right) \\ $$$${f}\left({x}\right)=−{xcos}\left({x}\right)+{sin}\left({x}\right) \\ $$$${j}_{\frac{\mathrm{3}}{\mathrm{2}}} \left({x}\right)=\frac{\mathrm{1}}{{x}\sqrt{{x}}}.\sqrt{\frac{\mathrm{2}}{\pi}}\left(−{xcos}\left({x}\right)+{sin}\left({x}\right)\right) \\ $$$$=\sqrt{\frac{\mathrm{2}}{\pi{x}}}.\left(\frac{{sin}\left({x}\right)}{{x}}−{cos}\left({x}\right)\right) \\ $$$$ \\ $$ | ||
Commented by TawaTawa last updated on 18/Feb/20 | ||
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$ | ||
Commented by mind is power last updated on 18/Feb/20 | ||
$${thanx}\:{miss}\:{you}\:{too}\: \\ $$ | ||