[mg14718] Multi-Variate Taylor Series Expansions

[Tom Bell]

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 tom...@stanford.edu, and thanks in advance for your
help.

----------------------------------------------------------------
Thomas (Tom) Bell
Gravity Probe-B, H.E.P.L. tom...@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)

[Carl Woll]

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