Re: Multi-Variate Taylor Series Expansions

*To*: mathgroup at smc.vnet.net*Subject*: [mg14732] Re: [mg14718] Multi-Variate Taylor Series Expansions*From*: Carl Woll <carlw at u.washington.edu>*Date*: Wed, 11 Nov 1998 17:53:37 -0500*Organization*: Physics Department, U of Washington*References*: <199811100621.BAA15036@smc.vnet.net.>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Tom, The Series function of Mathematica is intended to do multi-variate Taylor expansions, but it doesn't work right. If you look at the web page http://www.wolfram.com/support/Kernel/Symbols/System/Series.html you will find a way to augment the Series function to work better. I don't like the method given above, since I like to create series by adding O[x]^n to an expression, and the above method doesn't help here. Instead, one could modify the SeriesData function as follows: Unprotect[SeriesData]; SeriesData /: HoldPattern[SeriesData[a_,b_,c_,d__]]+HoldPattern[z:SeriesData[e_,__]] := SeriesData[a,b,c+z,d] /; a=!=e; Protect[SeriesData]; Then, in your example, I would do the following (F[x+dx,t+dt] + O[dx]^3) + O[dt]^3 The parenthesis above are necessary. This method should also work for your more complicated example. Note that using either of the above approaches will drop terms that are order O[dx]^3 or O[dt]^3, but not terms like dx^2 dt. Since your example has these terms dropped, you may want to do something different. Replace dx and dt by something like dx -> a de dt -> b de and do a series expansion on de: ser = (F[x+dx,t+dt]/.{dx->a de,dt->b de})+O[de]^3 and then fix things up by sending a and b back, as in Expand[Normal[ser]/.{a->dx/de,b->dt/de}] Good luck, Carl Woll Dept of Physics U of Washington Tom Bell wrote: > Is there a function in Mathematica that will do multi-variate Taylor > series > > expansions? For example, suppose I have > > function = F(x + dx, t + dt) > > then the expansion to second order about (x,t) should look something > like > > expansion = F(x,t) + dx D(F,x) + dt D(F,t) + (1/2) dx^2 D(F,{x,2}) + > > dx dt D(F,{x,t}) + (1/2) dt^2 D(F,{t,2}) + O(dx^3) + O(dt^3) > > The situation gets a little more complicated: the function may look like > > F(x + G(x + dx, t + dt), t + dt) and so on, so that the expansion should > be > > recursive. After expanding F, the function should keep going back and > expending G until no > > further expansions can be done. > > Please reply to tombell at stanford.edu, and thanks in advance for your > help. > > ---------------------------------------------------------------- > Thomas (Tom) Bell > Gravity Probe-B, H.E.P.L. tombell at stanford.edu > Stanford University 136D Escondido Village > Stanford, CA > 94305-4085 Stanford, CA 94305 650/725-6378 (o) > 650/497-4230 (h) 650/725-8312 (fax)

**References**:**Multi-Variate Taylor Series Expansions***From:*Tom Bell <tombell@stanford.edu>