# Kelvin function of the 1st kind

## Calculates the Kelvin functions of the first kind berv(x) and beiv(x), and their derivatives ber'v(x) and bei'v(x).

 order v real number x complex number
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Kelvin\ functions\\\hspace{60} of\ the\ 1st\ kind\ ber_\nu(x),\ bei_\nu(x)\\[10](1)\ x^2y''+xy'-(ix^2+\nu^2)y=0\\\hspace{25} y=c_1{ber_\nu(x)}+ic_2{bei_\nu(x)}\\[10](2)\ ber_\nu(x)+ibei_\nu(x)=J_\nu(x{\large e^{i\frac{3\pi}{4}}})\\\hspace{80}={\large(\frac{x}{2})^\nu e^{i\frac{3\nu\pi}{4}}\sum_{\small k=0}^{\small\infty}\frac{(\frac{ix^2}{4})^{k}}{k!\Gamma(\nu+k+1)}}\\[10](3)\ ber_\nu'(x)={\large\frac{ber_{\nu\tiny +1}(x)+bei_{\nu\tiny +1}(x)}{\sqrt{2}}}+{\large\frac{\nu}{x}}ber_\nu(x)\\\hspace{25} bei_\nu'(x)={\large\frac{bei_{\nu\tiny +1}(x)-ber_{\nu\tiny +1}(x)}{\sqrt{2}}}+{\large\frac{\nu}{x}}bei_\nu(x)\\$

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