# Bessel function Calculator

## Calculates the Bessel functions of the first kind Jv(x) and second kind Yv(x), and their derivatives J'v(x) and Y'v(x).

 order v real number x complex number
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Bessel\ functions\ of\\\ the\ 1st\ kind\ J_\nu(x)\ and\ 2nd\ kind\ Y_\nu(x)\\[10](1)\ x^2y''+xy'+(x^2-\nu^2)y=0\\\hspace{25} y=c_1J_\nu(x)+c_2Y_\nu(x)\\[10](2)\ J_\nu(x)={\large\sum_{\small k=0}^{\small\infty}\frac{(-1)^k}{k!\Gamma(k+\nu+1)}(\frac{x}{2})^{\nu+2k}}\\\hspace{30} Y_\nu(x)={\large\frac{J_\nu(x)cos(\nu\pi)-J_{-\nu}(x)}{sin(\nu\pi)}}\\[10](3)\ J'_\nu(x)=J_{\nu-1}(x)-{\large\frac{\nu}{x}}J_\nu(x)=-J_{\nu+1}(x)+{\large\frac{\nu}{x}}J_\nu(x)\\\hspace{25}Y'_\nu(x)=Y_{\nu-1}(x)-{\large\frac{\nu}{x}}Y_\nu(x)=-Y_{\nu+1}(x)+{\large\frac{\nu}{x}}Y_\nu(x)\\[10](4)\ {\large e^{\frac{x}{2}(t-{\large\frac{1}{t}})}}={\large\sum_{\small n=-\infty}^{\small\infty}}J_n(x)t^n,\hspace{20} n=integer\\$

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