Kelvin functions ber and bei - Maple Help

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