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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" ?2011-09-10
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    @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.2011-09-10

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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.2011-09-10
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    Whaddya mean "not so good"?! Did ya bother to read through the [succeeding sections](http://dlmf.nist.gov/10)?2011-09-10
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    I missed it, in hurry. I just focused on the linked page. I'm going back there. :)2011-09-10
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    There's also a bit about them at http://en.wikipedia.org/wiki/Kelvin_functions2011-09-10
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    [Wolfram Alpha's able to deal with 'em, by the way.](http://www.wolframalpha.com/input/?i=plot+KelvinKer%5BRange%5B0%2C+5%5D%2C+x%5D%2C+x+from+0+to+7)2011-09-10