Re: Questions about series and O[x]

*To*: mathgroup at smc.vnet.net*Subject*: [mg12797] Re: [mg12782] Questions about series and O[x]*From*: Daniel Lichtblau <danl>*Date*: Fri, 12 Jun 1998 04:05:29 -0400*References*: <199806100704.DAA13695@smc.vnet.net.>*Sender*: owner-wri-mathgroup at wolfram.com

Matthew D Litwin wrote: > > Hi, > > I'm working with series with fractional exponents, and have a few > questions on how to efficiently do certain operations. > > 1) Suppose you have a series like f=1 +x^(1/2) +x + x^(3/2) +O[x]^2 What > is the most efficient way to apply a transformation like x -> x^(2/3) > which would yield a series with order O[x]^(4/3). The best I've found > is to operate on Normal[f] and add the O[x] term by hand, but this > seems slower than it need be for long series. > > 2) Suppose you have two (or three) such series and are taking the > product, call it P, and you know that the only non-zero terms in P are > of the form A x^n, where n is an integer. Is there any way to avoid > computing the fractional terms in P? > > Thank you for any suggestions, > > -Matt Litwin > matt at math.ucsb.edu Not certain, but the answer to 2) is probably "no." For 1) you might do as follows. interleave[ll_List, len_Integer] := With[{zeroes=Table[0,{len}]}, Flatten[Map[{#,zeroes}&, ll]]] replaceWithPower[ser_SeriesData, (pow_Rational|pow_Integer)] := Module[ {g=ser, num=Numerator[pow]}, If [num!=1, g[[3]] = interleave[g[[3]], num-1]]; g[[4]] *= num; g[[5]] *= num; g[[6]] *= Denominator[pow]; g ] For example: In[34]:= InputForm[f = 1 + x^(1/2) + x + x^(3/2) + O[x]^2] Out[34]//InputForm= SeriesData[x, 0, {1, 1, 1, 1}, 0, 4, 2] In[35]:= g = replaceWithPower[f, 2/3] 1/3 2/3 4/3 Out[35]= 1 + x + x + x + O[x] (* Looks good, but let's check the InputForm to be certain *) In[36]:= InputForm[g] Out[36]//InputForm= SeriesData[x, 0, {1, 0, 1, 0, 1, 0, 1}, 0, 8, 6] To see why this works: In[37]:= ??SeriesData SeriesData[x, x0, {a0, a1, ... }, nmin, nmax, den] represents a power series in the variable x about the point x0. The ai are the coefficients in the power series. The powers of (x-x0) that appear are nmin/den, (nmin+1)/den, ... , nmax/den. Attributes[SeriesData] = {Protected, ReadProtected} Daniel Lichtblau Wolfram Research

**References**:**Questions about series and O[x]***From:*6500mdl0@ucsbuxa.ucsb.edu (Matthew D Litwin)