Modified Bessel Functions
Calculates the values of the first-kind modified Bessel functions Iv(x), the second-kind modified Bessel functions Kv(x), and their derivatives I'v(x), K'v(x).
The variable x accepts complex numbers as input.
| \(\normalsize Modified\ Bessel\ functions\ of\\ \ the\ 1st\ kind\ I_\nu(x)\ and\ 2nd\ kind\ K_\nu(x)\\ (1)\ x^2y''+xy'-(x^2+\nu^2)y=0\\ \hspace{25px} y=c_1I_\nu(x)+c_2K_\nu(x)\\ (2)\ I_\nu(x)={\large\displaystyle \sum_{\small k=0}^{\small\infty}\frac{1}{k!\Gamma(k+\nu+1)}(\frac{x}{2})^{\nu+2k}}\\ \hspace{25px} K_\nu(x)={\large\frac{\pi(I_{-\nu}(x)-I_\nu(x))}{2 \sin(\nu\pi)}}\\ (3)\ I'_\nu(x)=I_{\nu-1}(x)-{\large\frac{\nu}{x}}I_\nu(x)=I_{\nu+1}(x)+{\large\frac{\nu}{x}}I_\nu(x)\\ \hspace{25px}K'_\nu(x)=-K_{\nu-1}(x)-{\large\frac{\nu}{x}}K_\nu(x)=-K_{\nu+1}(x)+{\large\frac{\nu}{x}}K_\nu(x)\\ (4)\ {\large e^{\frac{x}{2}(t+{\large\frac{1}{t}})}}={\large\displaystyle \sum_{\small n=-\infty}^{\small\infty}}I_n(x)t^n,\hspace{20px} n=integer\\\) |
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