From: jos.bergervoet@nxp.com
On 19/05/25 4:06 PM, MM wrote:
> On Sat, 25 May 2019 10:14:13 +0100, mfreeman1943 wrote:
>
>> On Friday, May 24, 2019 at 3:29:19 PM UTC-7, MM wrote:
>>> Hello all.
>>> Straight to the point, stemming from a problem concerning a phi^3
>>> potential.
>>>
>>> How to find out the series expansion for s = s(t) given by the implicit
>>> equation:
>>>
>>> 18 t + s(1+s) (1+2s) =0
>>>
>>> There actually exists a closed expression for series s(t) = sum_k s_k
>>> t^k
>>
>> Since t is a cubic in s, you could use the solution of the cubic to get
>> s(t) - no series necessary.
>
> I need the series expansion, that's precisely the point. t> 0.
You could start in the usual way with Mathematica:
<0. Plot
them to be sure: *)
Plot[{s2}, {t,-2,2}] (* and the same for the others *)
(* Now let's expand in a point a>0, up to 7th power: *)
ser = Series[s2, {t,a, 7}]
(* or did you want the expansion to be around t=0 ? *)
ser = Series[s2, {t,0, 7}]
(* Anyhow, you can now ask for the series coefficients: *)
SeriesCoefficient[ser, 0]
SeriesCoefficient[ser, 1]
(* ... *)
SeriesCoefficient[ser, 6]
SeriesCoefficient[ser, 7]
>
> I don't see any obvious way to find out the generic s_k .
If you want a closed form expression for any k then the above
approach does not immediately help..
And if you want at least to have reasonably short expressions,
you will have to stick to the a=0 case. For general a, the 7th
coefficient, for (t-a)^7, spans more than 1000 lines of text,
whereas the 6th already needs a few pages:
2/3
Coeff[6]= (6428310336 3 (1 + I Sqrt[3])
3
(-245268 Sqrt[3] a + 69536102400 Sqrt[3] a +
5 7
5996269108297728 Sqrt[3] a - 900884751948397412352 Sqrt[3] a +
9
14010559662301476556898304 Sqrt[3] a +
11
301696925355848164725529509888 Sqrt[3] a +
2 2 2
143 Sqrt[-1 + 34992 a ] - 338162688 a Sqrt[-1 + 34992 a ] +
4 2
3565569466368 a Sqrt[-1 + 34992 a ] +
6 2
5632312076917014528 a Sqrt[-1 + 34992 a ] -
8 2
169643528674590528110592 a Sqrt[-1 + 34992 a ] -
10 2
2793490049591186710421569536 a Sqrt[-1 + 34992 a ])) /
2 11/2 2 19/3
((-1 + 34992 a ) (-324 a + Sqrt[3] Sqrt[-1 + 34992 a ]) ) -
1/3
(6428310336 3 (1 - I Sqrt[3])
3
(-59940 Sqrt[3] a + 91409741568 Sqrt[3] a -
5 7
4491912250146816 Sqrt[3] a - 351824814370380054528 Sqrt[3] a +
9
13222983823296363969380352 Sqrt[3] a +
11
63820503440660188691938934784 Sqrt[3] a -
2 2 2
143 Sqrt[-1 + 34992 a ] - 255161664 a Sqrt[-1 + 34992 a ] +
4 2
46332812021760 a Sqrt[-1 + 34992 a ] +
6 2
1327185513742860288 a Sqrt[-1 + 34992 a ] -
8 2
130878830945781380284416 a Sqrt[-1 + 34992 a ] -
10 2
590930587413520265666101248 a Sqrt[-1 + 34992 a ])) /
2 11/2 2 17/3
((-1 + 34992 a ) (-324 a + Sqrt[3] Sqrt[-1 + 34992 a ]) )
--
Jos
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