Hankel Functions
First-kind Hankel functions Hv(1)(x), second-kind Hankel functions Hv (2)(x), and their derivatives H'v(1)(x), H'v(2)(x).
The variable x accepts complex numbers.
| \(\normalsize Hankel\ functions\ of\ the\ 1st\ kind\\ \hspace{75px} H^{(1)}_\nu(x)\ and\ 2nd\ kind\ H^{(2)}_\nu(x)\\ (1)\ x^2y''+xy'+(x^2-\nu^2)y=0\\ \hspace{25px} y=c_1H^{(1)}_\nu(x)+c_2H^{(2)}_\nu(x)\\ (2)\ H^{(1)}_\nu(x)=J_\nu(x)+iY_\nu(x)\\ \hspace{25px} H^{(2)}_\nu(x)=J_\nu(x)-iY_\nu(x)\\ (3)\ H'^{(1)}_\nu(x)=H^{(1)}_{\nu-1}(x)-{\large\frac{\nu}{x}}H^{(1)}_\nu(x)=-H^{(1)}_{\nu+1}(x)+{\large\frac{\nu}{x}}H^{(1)}_\nu(x)\\ \hspace{25px}H'^{(2)}_\nu(x)=H^{(2)}_{\nu-1}(x)-{\large\frac{\nu}{x}}H^{(2)}_\nu(x)=-H^{(2)}_{\nu+1}(x)+{\large\frac{\nu}{x}}H^{(2)}_\nu(x)\\\) |
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