# Modified spherical Bessel function Calculator

## Calculates the modified spherical Bessel functions of the first kind iv(x) and the second kind kv(x), and their derivatives i'v(x) and k'v(x).

 order v real number x complex number 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Modified\ spherical\ Bessel\ functions\\\ of\ the\ 1st\ kind\ i_\nu(x)\ and\ 2nd\ kind\ k_\nu(x)\\[10](1)\ x^2w''+2xw'-(x^2+\nu(\nu+1))w=0\\\hspace{25} w=c_1i_\nu(x)+c_2k_\nu(x)\\[10](2)\ i_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}I_{\nu+\frac{1}{2}}(x)\\\hspace{25}k_\nu(x)={\large\sqrt{\frac{2}{\pi x}}}K_{\nu+\frac{1}{2}}(x)\\[10](3)\ i'_\nu(x)=i_{\nu-1}(x)-{\large\frac{\nu+1}{x}}i_\nu(x)=-i_{\nu+1}(x)+{\large\frac{\nu}{x}}i_\nu(x)\\\hspace{25}k'_\nu(x)=-k_{\nu-1}(x)-{\large\frac{\nu-1}{x}}k_\nu(x)=-k_{\nu+1}(x)+{\large\frac{\nu}{x}}k_\nu(x)\\$

Modified spherical Bessel function
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