# Kelvin function of the 3rd kind Calculator

## Calculates the Kelvin functions of the third kind herv(x) and heiv(x), and their derivatives her'v(x) and hei'v(x).

 order v real number x complex number
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Kelvin\ functions\\\hspace{60} of\ the\ 3rd\ kind\qquad her_\nu(x),\ hei_\nu(x)\\[10](1)\ x^2y''+xy'-(ix^2+\nu^2)y=0\\\hspace{25} y=c_1{her_\nu(x)}+ic_2{hei_\nu(x)}\\[10](2)\ her_\nu(x)+ihei_\nu(x)=H^{(1)}_\nu(x{\large e^{i\frac{3\pi}{4}}})\\\hspace{135}={\large\frac{2}{i\pi}}\left(ker_\nu(x)+ikei_\nu(x)\right)\\[10](3)\ her_\nu'(x)={\large\frac{her_{\nu\tiny +1}(x)+hei_{\nu\tiny +1}(x)}{\sqrt{2}}} +{\large\frac{\nu}{x}}her_\nu(x)\\\hspace{25}hei_\nu'(x)={\large\frac{hei_{\nu\tiny +1}(x)-her_{\nu\tiny +1}(x)}{\sqrt{2}}} +{\large\frac{\nu}{x}}hei_\nu(x)\\$

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