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## Posts Tagged ‘Expansion’

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? ðŸ˜›

#### 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. ðŸ˜‰

#### 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 ðŸ™‚
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