Spherical Hankel functions
First-kind spherical Hankel functions hv(1)(x), second-kind spherical Hankel functions hv (2)(x) and their derivatives h'v(1)(x), h'v(2)(x).
The variable x accepts complex numbers as input.
| \(\normalsize Spherical\ Hankel\ functions\ of\\ \ the\ 1st\ kind\ h^{(1)}_\nu(x)\ and\ 2nd\ kind\ h^{(2)}_\nu(x)\\ (1)\ x^2w''+2xw'+(x^2-\nu(\nu+1))w=0\\ \hspace{25px} w=c_1h^{(1)}_\nu(x)+c_2h^{(2)}_\nu(x)\\ (2)\ h^{(1)}_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}H^{(1)}_{\nu+\frac{1}{2}}(x)=j_\nu(x)+iy_\nu(x)\\ \hspace{25px}h^{(2)}_\nu(x)=j_\nu(x)-iy_\nu(x)\\ (3)\ h'^{(1)}_\nu(x)=h^{(1)}_{\nu-1}(x)-{\large\frac{\nu+1}{x}}h^{(1)}_\nu(x)=-h^{(1)}_{\nu+1}(x)+{\large\frac{\nu}{x}}h^{(1)}_\nu(x)\\ \hspace{25px}h'^{(2)}_\nu(x)=h^{(2)}_{\nu-1}(x)-{\large\frac{\nu+1}{x}}h^{(2)}_\nu(x)=-h^{(2)}_{\nu+1}(x)+{\large\frac{\nu}{x}}h^{(2)}_\nu(x)\\\) |
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