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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.

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@guarav Could you give the author(s) of the book, please. Is it published in English ? Is it, maybe, a chapter in the book of Rajput titled "Mathematical Physics" ? – Sasha Sep 10 '11 at 14:03
@Sasha Is this book [Mathematical Physics by B.S. Rajput] so famous? I've it and I am learning ordinary differential equations with two books, Advanced Differential Equations by M.D. Raisinghania and Mathematical Physics by B.S. Rajput. – gaurav Sep 10 '11 at 14:22
up vote 3 down vote accepted

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.

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Viewed both links. First one is not so good, but functions.wolfram is useful. When I came back to my question, you edited your answer -- which is more effective, because I found similar expression in the book. Thanks. – gaurav Sep 10 '11 at 12:47
Whaddya mean "not so good"?! Did ya bother to read through the succeeding sections? – J. M. Sep 10 '11 at 12:50
I missed it, in hurry. I just focused on the linked page. I'm going back there. :) – gaurav Sep 10 '11 at 12:55
There's also a bit about them at – Gerry Myerson Sep 10 '11 at 12:56

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