Modified spherical Bessel functions
Calculates the values of the first-kind modified spherical Bessel functions iv(x), the second-kind modified spherical Bessel functions kv (x) and their derivatives i'v(x) and k'v(x).
The variable x accepts complex numbers as input.
| \(\normalsize Modified\ spherical\ Bessel\ functions\\ \ of\ the\ 1st\ kind\ i_\nu(x)\ and\ 2nd\ kind\ k_\nu(x)\\ (1)\ x^2w''+2xw'-(x^2+\nu(\nu+1))w=0\\ \hspace{25px} w=c_1i_\nu(x)+c_2k_\nu(x)\\ (2)\ i_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}I_{\nu+\frac{1}{2}}(x)\\ \hspace{25px}k_\nu(x)={\large\sqrt{\frac{2}{\pi x}}}K_{\nu+\frac{1}{2}}(x)\\ (3)\ i'_\nu(x)=i_{\nu-1}(x)-{\large\frac{\nu+1}{x}}i_\nu(x)=-i_{\nu+1}(x)+{\large\frac{\nu}{x}}i_\nu(x)\\ \hspace{25px}k'_\nu(x)=-k_{\nu-1}(x)-{\large\frac{\nu-1}{x}}k_\nu(x)=-k_{\nu+1}(x)+{\large\frac{\nu}{x}}k_\nu(x)\\\) |
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