The Kelvin functions (sometimes known as the Thomson functions) are defined by the following equations:

The Kelvin functions are all real valued for real x and positive v.

$\mathrm{KelvinBer}\left(0\,0\right)$

$1$

$\mathrm{KelvinKei}\left(1.5-\mathrm{I}\,2.6+3\mathrm{I}\right)$

$\mathrm{-0.08160376508}-0.03651099032\mathrm{I}$

$\mathrm{series}\left(\mathrm{KelvinHer}\left(1\,x\right)\,x\,3\right)$

$-\frac{\sqrt{2}}{\mathrm{\pi }}{x}^{\mathrm{-1}}+\frac{1}{4}\frac{\sqrt{2}\left(-\ln \left(\left(\mathrm{-1}-\mathrm{I}\right)x\right)-\ln \left(\left(\mathrm{-1}+\mathrm{I}\right)x\right)-2\mathrm{\gamma }+3\ln \left(2\right)+\mathrm{I}\ln \left(\left(\mathrm{-1}-\mathrm{I}\right)x\right)-\mathrm{I}\ln \left(\left(\mathrm{-1}+\mathrm{I}\right)x\right)+1-\mathrm{\pi }\right)}{\mathrm{\pi }}x+O\left(x^3\right)$

$\mathrm{convert}\left(\mathrm{KelvinBei}\left(v\,x\right)\,\mathrm{BesselJ}\right)$

$\frac{\mathrm{I}}{2}\left(\mathrm{BesselJ}\left(v\,\left(-\frac{1}{2}-\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)-\mathrm{BesselJ}\left(v\,\left(-\frac{1}{2}+\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)\right)$

$\mathrm{diff}\left(\mathrm{KelvinHei}\left(v\,x\right)\,x\right)$

$\frac{\sqrt{2}\left(\mathrm{KelvinHei}\left(v+1\,x\right)-\mathrm{KelvinHer}\left(v+1\,x\right)\right)}{2}+\frac{v\mathrm{KelvinHei}\left(v\,x\right)}{x}$

$\mathrm{convert}\left(\mathrm{KelvinBer}\left(v\,x\right)\,\mathrm{BesselJ}\right)$

$\frac{\mathrm{BesselJ}\left(v\,\left(-\frac{1}{2}-\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)}{2}+\frac{\mathrm{BesselJ}\left(v\,\left(-\frac{1}{2}+\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)}{2}$

$\mathrm{convert}\left(\mathrm{KelvinBei}\left(v\,x\right)\,\mathrm{Bessel}\right)$

$\frac{\mathrm{I}}{2}\left(\mathrm{BesselJ}\left(v\,\left(-\frac{1}{2}-\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)-\mathrm{BesselJ}\left(v\,\left(-\frac{1}{2}+\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)\right)$

$\mathrm{convert}\left(\mathrm{KelvinKer}\left(v\,x\right)\,\mathrm{BesselK}\right)$

$\frac{\mathrm{BesselK}\left(v\,\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)+{\left({\ⅇ}^{\frac{\mathrm{I}}{2}v\mathrm{\pi }}\right)}^2\mathrm{BesselK}\left(v\,\left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)}{2{\ⅇ}^{\frac{\mathrm{I}}{2}v\mathrm{\pi }}}$

$\mathrm{convert}\left(\mathrm{KelvinHer}\left(v\,x\right)\,\mathrm{Hankel}\right)$

$\frac{\mathrm{HankelH1}\left(v\,\left(-\frac{1}{2}+\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)}{2}+\frac{\mathrm{HankelH2}\left(v\,\left(-\frac{1}{2}-\frac{\mathrm{I}}{2}\right)x\sqrt{2}\right)}{2}$

Abramowitz, M., and Stegun, I. Handbook of Mathematical Functions, Section 9.9. Washington: National Bureau of Standards Applied Mathematics, 1964.

Erdelyi, A., ed. Higher Transcendental Functions, Section 7.2.3. New York: McGraw-Hill, 1953.