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Question Number 82073 by TawaTawa last updated on 18/Feb/20

$$\mathrm{Show}\mathrm{that}:{\mathrm{j}}_{3/2}\left(\mathrm{x}\right)=\frac{\sqrt{2}}{\pi \mathrm{x}}(\frac{\sin \mathrm{x}}{\mathrm{x}}-\cos \mathrm{x})$$

Commented by jagoll last updated on 18/Feb/20

$$whatis{j}_{\frac{3}{2}}?$$

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

$$J_a\left(x\right)={\left(\frac{x}{2}\right)}^a\sum _{k⩾0}\frac{{(-1)}^k{x}^{2k}}{2^{2k}k!.\mathrm{\Gamma }(k+1+a)}$$$$j_{\frac{3}{2}}\left(x\right)={\left(\frac{x}{2}\right)}^{\frac{3}{2}}\sum _{k⩾0}\frac{{(-1)}^k{x}^{2k}}{2^{2k}k!\mathrm{\Gamma }(k+1+\frac{3}{2})}$$$$\mathrm{\Gamma }(k+1+\frac{3}{2})=(k+\frac{3}{2})(k+\frac{1}{2})........\frac{1}{2}.\sqrt{\pi }$$$$=\frac{(2k+3).....(2k+1).....\sqrt{\pi }}{2^{k+2}}$$$$j_{\frac{3}{2}}\left(x\right)=\frac{1}{x\sqrt{x}}\frac{\sqrt{2}}{\sqrt{\pi }}\sum _{k⩾0}\frac{{(-1)}^k{x}^{2k+3}}{(2k+3).(2k+1)!}$$$$f\left(x\right)=\sum _{k⩾0}\frac{{(-1)}^k{x}^{2k+3}}{(2k+3)(2k+1)!}$$$$f^{\prime }\left(x\right)=\sum _{k⩾0}\frac{{(-1)}^k{x}^{2k+2}}{(2k+1)!}=xsin\left(x\right)$$$$f\left(x\right)=-xcos\left(x\right)+sin\left(x\right)$$$$j_{\frac{3}{2}}\left(x\right)=\frac{1}{x\sqrt{x}}.\sqrt{\frac{2}{\pi }}(-xcos\left(x\right)+sin\left(x\right))$$$$=\sqrt{\frac{2}{\pi x}}.(\frac{sin\left(x\right)}{x}-cos\left(x\right))$$$$$$

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

$$thanxmissyoutoo$$

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