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\makeatletter












\makeatother


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\begin{document}
\pagestyle{empty}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
{\newpage
\clearpage
\samepage \( \frac{\partial ^{2}R}{\partial x^{2}}+\frac{1}{x}\frac{\partial R}{\partial x}+(1-\frac{\nu ^{2}}{x^{2}})R=0 \)
}

{\newpage
\clearpage
\samepage \( x^{\alpha }\sum _{j=0}^{\infty }a_{j}x^{j} \)
}

{\newpage
\clearpage
\samepage \( a_{0}\neq 0 \)
}

{\newpage
\clearpage
\samepage \( \nu ^{2}=\alpha ^{2} \)
}

{\newpage
\clearpage
\samepage \( a_{j}=-\frac{1}{j(j+2\alpha )} \)
}

{\newpage
\clearpage
\samepage \( a_{1}=0 \)
}

{\newpage
\clearpage
\samepage \( a_{2j}=\frac{(-1)^{j}a_{0}\Gamma (\alpha +1)}{2^{2j}j!\Gamma (j+\alpha +1)} \)
}

{\newpage
\clearpage
\samepage \( \Gamma (z) \)
}

{\newpage
\clearpage
\samepage \( a_{0}=(2^{\gamma }\Gamma (\alpha +1))^{-1} \)
}

{\newpage
\clearpage
\samepage \( J_{\nu }(x)=(\frac{x}{2})^{\nu }\sum _{j=0}^{\infty }(-1)^{j}\frac{1}{\Gamma (j+1)\Gamma (\nu +j+1)}(\frac{x}{2})^{2j} \)
}

{\newpage
\clearpage
\samepage \( J_{\nu }(x)=(\frac{x}{2})^{-\nu }\sum _{j=0}^{\infty }(-1)^{j}\frac{1}{\Gamma (j+1)\Gamma (j-\nu +1)}(\frac{x}{2})^{2j} \)
}

{\newpage
\clearpage
\samepage \( \nu =n \)
}

{\newpage
\clearpage
\samepage \( J_{-n}(x)=(-1)^{n}(\frac{x}{2})^{n}\sum _{j=0}^{\infty }(-1)^{j}\frac{1}{\Gamma (j+1)\Gamma (j+n+1)}(\frac{x}{2})^{2j} \)
}

{\newpage
\clearpage
\samepage \( =(-1)^{n}J_{n}(x) \)
}

{\newpage
\clearpage
\samepage \( \nu  \)
}

{\newpage
\clearpage
\samepage \( N_{\nu }(x)=\frac{1}{\pi }[\frac{\partial J_{\nu }}{\partial \nu }-(-1)^{\nu }\frac{\partial J_{-\nu }}{\partial \nu }] \)
}

{\newpage
\clearpage
\samepage \( N_{\nu }=\frac{cos(\nu \pi )J_{\nu }(x)-J_{-\nu }(x)}{sin(\nu \pi )} \)
}

{\newpage
\clearpage
\samepage \( J_{v},N_{\nu } \)
}

{\newpage
\clearpage
\samepage \( J_{\nu }(x)\rightarrow \frac{1}{\Gamma (\nu +1)}(\frac{x}{2})^{\nu } \)
}

{\newpage
\clearpage
\samepage \(>\)
}

{\newpage
\clearpage
\samepage \( N_{\nu }(x)\rightarrow \{\begin{array}{cc}
\frac{2}{\pi }(ln(\frac{x}{2})+\gamma ) & \nu =0\\ 
\frac{-\Gamma (\nu )}{\pi }(\frac{2}{x})^{\nu } & \nu >0
\end{array}
 \)
}

{\newpage
\clearpage
\samepage \( J_{\nu }(x)\sim \sqrt{\frac{2}{\pi x}}cos(x-\frac{\nu \pi }{2}-\frac{\pi }{4}) \)
}

{\newpage
\clearpage
\samepage \( N_{\nu }(x)\sim \sqrt{\frac{2}{\pi x}}sin(x-\frac{\nu \pi }{2}-\frac{\pi }{4}) \)
}

{\newpage
\clearpage
\samepage \( J_{\nu -1}+J_{\nu +1}=\frac{2\nu }{x}J_{\nu } \)
}

{\newpage
\clearpage
\samepage \( e^{\frac{x}{2}(t-\frac{1}{t})}=\sum _{n=-\infty }^{\infty }J_{n}(x)t^{n} \)
}

{\newpage
\clearpage
\samepage \( e^{xsin(\theta )} \)
}

{\newpage
\clearpage
\samepage \( \int _{0}^{a}d\rho *\rho J_{\nu }(k_{1}\rho )J_{\nu }(k_{2}\rho )=\frac{a^{2}}{2}J^{2}_{\nu +1}(x_{\nu n})\delta _{k_{1}k_{2}} \)
}

{\newpage
\clearpage
\samepage \( x_{\nu n} \)
}

{\newpage
\clearpage
\samepage \( J_{\nu } \)
}

{\newpage
\clearpage
\samepage \( k_{i}a \)
}

{\newpage
\clearpage
\samepage \( k_{i}a= \)
}

{\newpage
\clearpage
\samepage \( J_{\nu }' \)
}

{\newpage
\clearpage
\samepage \( f(\rho )=\sum _{n=1}^{\infty }c_{\nu n}J_{\nu }(x_{\nu n}\frac{\rho }{a}) \)
}

{\newpage
\clearpage
\samepage \( c_{\nu n}=\frac{2}{a^{2}J_{\nu +1}^{2}(x_{\nu n})}\int _{0}^{a}d\rho *\rho f(\rho )J_{\nu }(x_{\nu n}\frac{\rho }{a}) \)
}


\end{document}
