# Modified Bessel function Calculator

## Calculates the modified 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
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Modified\ Bessel\ functions\ of\\\ the\ 1st\ kind\ I_\nu(x)\ and\ 2nd\ kind\ K_\nu(x)\\[10](1)\ x^2y''+xy'-(x^2+\nu^2)y=0\\\hspace{25} y=c_1I_\nu(x)+c_2K_\nu(x)\\[10](2)\ I_\nu(x)={\large\sum_{\small k=0}^{\small\infty}\frac{1}{k!\Gamma(k+\nu+1)}(\frac{x}{2})^{\nu+2k}}\\\hspace{25} K_\nu(x)={\large\frac{\pi(I_{-\nu}(x)-I_\nu(x))}{2sin(\nu\pi)}}\\[10](3)\ I'_\nu(x)=I_{\nu-1}(x)-{\large\frac{\nu}{x}}I_\nu(x)=I_{\nu+1}(x)+{\large\frac{\nu}{x}}I_\nu(x)\\\hspace{25}K'_\nu(x)=-K_{\nu-1}(x)-{\large\frac{\nu}{x}}K_\nu(x)=-K_{\nu+1}(x)+{\large\frac{\nu}{x}}K_\nu(x)\\[10](4)\ {\large e^{\frac{x}{2}(t+{\large\frac{1}{t}})}}={\large\sum_{\small n=-\infty}^{\small\infty}}I_n(x)t^n,\hspace{20} n=integer\\$

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