What are the $\ker$, $\mathrm{kei}$ functions? In a book titled 'Ordinary Differential Equations and Useful Polynomials', under the chapter 'Bessel's function', the author has introduced four new functions $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, $\mathrm{kei}$ saying that
$\mathrm{ber}$, $\mathrm{bei}$ are Bessel real and Bessel imaginary functions. $\ker$, $\mathrm{kei}$ are their analogues.
The book provides enough information about $\mathrm{ber}$ and $\mathrm{bei}$, but there is not enough material about $\ker$ and $\mathrm{kei}$!
(or, if there is, I'm not able to understand.)
I completely understand about $\mathrm{ber}$/$\mathrm{bei}$ but concept is still not clear about $\ker$/$\mathrm{kei}$.
Any online reference is appreciated.

They're the Kelvin functions. There's quite a bit about them at the DLMF and the Wolfram Functions site.
Briefly, the Kelvin functions $\mathrm{ker}_n(x)$ and $\mathrm{kei}_n(x)$ satisfy the following relationship with the modified Bessel function of the second kind $K_n(x)$:
$$\mathrm{ker}_n(x)+i\,\mathrm{kei}_n(x)=\exp(-i\pi n/2)K_n(x\exp(i\pi/4))$$
where $x$ is positive.
