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KelvinBer, KelvinBei

Kelvin functions ber and bei

KelvinKer, KelvinKei

Kelvin functions ker and kei

KelvinHer, KelvinHei

Kelvin functions her and hei

 Calling Sequence KelvinBer(v, x) KelvinBei(v, x) KelvinKer(v, x) KelvinKei(v, x) KelvinHer(v, x) KelvinHei(v, x)

Parameters

 v - algebraic expression (the order or index) x - algebraic expression (the argument)

Description

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

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

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

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

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

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

$\mathrm{KelvinHer}\left(v,x\right)-I\mathrm{KelvinHei}\left(v,x\right)=\mathrm{HankelH2}\left(v,x\left(-\frac{1}{2}\sqrt{2}-\frac{1}{2}I\sqrt{2}\right)\right)$

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

Examples

 > $\mathrm{KelvinBer}\left(0,0\right)$
 ${1}$ (1)
 > $\mathrm{KelvinKei}\left(1.5-I,2.6+3I\right)$
 ${-0.08160376508}{-}{0.03651099032}{}{I}$ (2)
 > $\mathrm{series}\left(\mathrm{KelvinHer}\left(1,x\right),x,3\right)$
 ${-}\frac{\sqrt{{2}}}{{\mathrm{\pi }}}{}{{x}}^{{-1}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}{}\left({-}{\mathrm{ln}}{}\left(\left({-1}{-}{I}\right){}{x}\right){-}{\mathrm{ln}}{}\left(\left({-1}{+}{I}\right){}{x}\right){-}{2}{}{\mathrm{\gamma }}{+}{3}{}{\mathrm{ln}}{}\left({2}\right){+}{I}{}{\mathrm{ln}}{}\left(\left({-1}{-}{I}\right){}{x}\right){-}{I}{}{\mathrm{ln}}{}\left(\left({-1}{+}{I}\right){}{x}\right){+}{1}{-}{\mathrm{\pi }}\right)}{{\mathrm{\pi }}}{}{x}{+}{O}{}\left({{x}}^{{3}}\right)$ (3)
 > $\mathrm{convert}\left(\mathrm{KelvinBei}\left(v,x\right),\mathrm{BesselJ}\right)$
 $\frac{{I}}{{2}}{}\left({\mathrm{BesselJ}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right){-}{\mathrm{BesselJ}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)\right)$ (4)
 > $\frac{\partial }{\partial x}\mathrm{KelvinHei}\left(v,x\right)$
 $\frac{\sqrt{{2}}{}\left({\mathrm{KelvinHei}}{}\left({v}{+}{1}{,}{x}\right){-}{\mathrm{KelvinHer}}{}\left({v}{+}{1}{,}{x}\right)\right)}{{2}}{+}\frac{{v}{}{\mathrm{KelvinHei}}{}\left({v}{,}{x}\right)}{{x}}$ (5)
 > $\mathrm{convert}\left(\mathrm{KelvinBer}\left(v,x\right),\mathrm{BesselJ}\right)$
 $\frac{{\mathrm{BesselJ}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)}{{2}}{+}\frac{{\mathrm{BesselJ}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)}{{2}}$ (6)
 > $\mathrm{convert}\left(\mathrm{KelvinBei}\left(v,x\right),\mathrm{Bessel}\right)$
 $\frac{{I}}{{2}}{}\left({\mathrm{BesselJ}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right){-}{\mathrm{BesselJ}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)\right)$ (7)
 > $\mathrm{convert}\left(\mathrm{KelvinKer}\left(v,x\right),\mathrm{BesselK}\right)$
 $\frac{{\mathrm{BesselK}}{}\left({v}{,}\left(\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right){+}{\left({{ⅇ}}^{\frac{{I}}{{2}}{}{v}{}{\mathrm{\pi }}}\right)}^{{2}}{}{\mathrm{BesselK}}{}\left({v}{,}\left(\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)}{{2}{}{{ⅇ}}^{\frac{{I}}{{2}}{}{v}{}{\mathrm{\pi }}}}$ (8)
 > $\mathrm{convert}\left(\mathrm{KelvinHer}\left(v,x\right),\mathrm{Hankel}\right)$
 $\frac{{\mathrm{HankelH1}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)}{{2}}{+}\frac{{\mathrm{HankelH2}}{}\left({v}{,}\left({-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\right){}{x}{}\sqrt{{2}}\right)}{{2}}$ (9)

References

 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.