球ベッセル関数

変数xは複素数入力可能です。

\(\normalsize Spherical\ Bessel\ function\ of\\ \ the\ 1st\ kind\ j_\nu(x)\ and\ 2nd\ kind\ y_\nu(x)\\ (1)\ x^2w''+2xw'+(x^2-\nu(\nu+1))w=0\\ \hspace{25px} w=c_1j_\nu(x)+c_2y_\nu(x)\\ (2)\ j_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}J_{\nu+\frac{1}{2}}(x)\\ \hspace{25px}y_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}Y_{\nu+\frac{1}{2}}(x)\\ (3)\ j'_\nu(x)=j_{\nu-1}(x)-{\large\frac{\nu+1}{x}}j_\nu(x)=-j_{\nu+1}(x)+{\large\frac{\nu}{x}}j_\nu(x)\\ \hspace{25px}y'_\nu(x)=y_{\nu-1}(x)-{\large\frac{\nu+1}{x}}y_\nu(x)=-y_{\nu+1}(x)+{\large\frac{\nu}{x}}y_\nu(x)\\\)