Kelvin functions: ber, bei, ber', bei'

[electricpete]

"Magnetic Fields" by Koepfel has an exact equation to compute ac resistances accounting for skin effects in single solid conductors.

It includes some functions called Kelvin functions which are relatives of the bessel function. Specifically, it includes the four functions: ber, bei, ber' and bei'

I think ber' is the derivative of ber, and similar for bei'

There is some discussion of ber and bei here including closed form expression based on Bessel functions:

http://mathworld.wolfram.com/Ber.html

But nothing about ber' and bei'

Does anyone have any idea how to compute these ber' and bei' ?

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[IRstuff]

derivatives:

http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php#ga25

TTFN

FAQ731-376

 

[IDS]

IRStuff - just a word of thanks for the link to the Codecogs site. I hadn't seen it before, and it looks really useful.

Thanks.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[electricpete]

Thanks for the help.

It looks like it will be complicated no matter how I do it.

Start with bei from IRStuff's link.
First thing I have to decide if the argument is below MIN. What value to use for MIN?

If below MIN, I have to do a sum for n>=0. How far should I go before truncating the series?

If above MIN, now I have another expression involving new functions f, g, and ker. f and g are more unbounded series. ker is yet another function. Let's follow the trail...

ker again has a brnach depending on min. These expressions are even more complicated involving beta, Hn, f(x), g(x), and ber and bei! But calling back to bei would put me in an infinite loop since I started out calculating bei. Let's follow the trail to the another argument ber....

ber has a structure similar to bei. Except if we follow the branch x>Min, we are led to an equation bei=... under the heading ber! Maybe it's just a typographical error and should have been ber=.... ? I'm getting a headache and it's not from too much partying last night!

So now I looked back at my own link. They have closed form expression for ber and bei from bessel functions. I figure I can calculate the derivative using 5-point formula with careful selection of delta-x to balance roundoff error against truncation error. But when I look closely, I see the arguments to I0 and J0 involve things like (-1)^(1/4) = exp(j*Pi/4) = complex number. But excels I0 and J0 take real arguments. Another dead end? Does anyone know how to address that?

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[electricpete]

I have my problem solved. Rather than fighting to compute the functions myself, I just had to look around my computer for other programs that already have them built in.

Maple has ber and bei built in. Also the ber' and bei' are obtained simply by requesting Maple to provide the derivatives of ber and bei as shown in attached.

Thanks again.

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