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Two days ago I was trying to expand in series a lot of functions… so I ask myself, Could it be done in SAGE? It should be possible… but, How? 😛

So I ask on google group.

Solution by Andrzej Chrzeszczyk

sage: var('r');
        sage: f=2*r/sinh(2*r)
        sage: f.taylor(r,0,5)
        14/45*r^4 - 2/3*r^2 + 1
        sage: maxima(f).powerseries('r',0)
        -4*r*'sum((2^(2*i2-1)-1)*2^(2*i2-1)*bern(2*i2)*r^(2*i2-1)/
        (2*i2)!,i2,0,inf)

This solution uses a power series expansion from maxima… really nice feature! Isn’t it?
Ah… and this expansion is around r=0.

If one would like the asymptotic expansion r\to\infty,

sage: maxima(f).powerseries('r',infinity)
        -4*r*'sum((2^(2*i3-1)-1)*2^(2*i3-1)*bern(2*i3)*r^(2*i3-1)/
        (2*i3)!,i3,0,inf)

However, note that this expansion coincides with the previous one, i.e., it’s the function itself. It couldn’t be that perfect. 😉

Solution by Francois Maltey

Use the Taylor command of SAGE,

  • Around zero
    sage: taylor (2*x/sinh(2*x), x, 0, 10)
            -292/13365*x^10 + 254/4725*x^8 - 124/945*x^6 + 14/45*x^4 - 2/3*x^2 + 1
  • Around infinity… a trick! change x\mapsto 1/x and expand around zero 🙂
    sage: taylor (2*1/x/((exp(2/x)-exp(-2/x))/2), x, 0, 12)
            4*e^(-10/x)/x + 4*e^(-6/x)/x + 4*e^(-2/x)/x
  • Thank you guys!

    Enjoy!

     

    Dox

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