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MathGroup Archive 2002

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Re: Taylor Series Expansions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg32382] Re: [mg32372] Taylor Series Expansions
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 17 Jan 2002 02:23:36 -0500 (EST)
  • References: <200201160831.DAA09716@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Joe Helfand wrote:
> 
> Wow!
> 
>     I have definitely come to the right place.  Thanks for all the
> responses.  Using the Map built in function solved my problem (it still
> took a bit, so you can imagine what I was dealing with).  Here is
> something else which I have wasted some time on not knowing as much
> about Mathematica as I should.  It has to do with multi-variable Taylor
> series expansion.  Mathematica has a built in Series function.  But when
> you use this for multi-variable functions, it doesn't do quite what I'd
> expect.  Let's say I have a function for two fariables, and I want to
> expand to 2nd order.  When I use Series, it expands each varible to
> second order, but includes the cross terms, which I want to belong to a
> 4th order expansion.  For example:
> 
> In[1172]:=
> Normal[Series[Exp[x y], {x, 0, 2}, {y, 0, 2}]]
> 
> Out[1172]=
> \!\(1 + x\ y + \(x\^2\ y\^2\)\/2\)
> 
> But what I really want is just 1 + x y, where if I go to fourth order,
> then I'll take the x^2 y^2 / 2.  I had to take some time to write some
> sloppy Taylor series expansion functions that did what I wanted.  Is
> there a way to get around this problem or do you have any suggestions?
> 
> Thanks Again,
> Joe

The best way I have seen was posted a few years ago:

http://library.wolfram.com/mathgroup/archive/1998/Jul/msg00058.html

If you want to expand somewhere other than the origin you will have to
modify the code a bit. An alternative method that works readily for this
case may be found at:

http://library.wolfram.com/mathgroup/archive/1998/Jun/msg00332.html


On the topic of Integrate and sums, it has been pointed out that current
behavior (of built-in Integrate) is NOT to map over the sum because we
may then lose track of "bad" things (singularities, say) that could
cancel. We are experimenting with changes that do the mapping and check
after the fact for presence of bad stuff. If nothing obvious breaks then
we will do it this way in our next release and you can get results like
the one below.

integrand = (a*c*Cos[t])/(g*s) + (b*q*Cos[2*t])/(c*f) +
(c*Cos[3*t])/(a*d) + 
 (d*f*Cos[4*t])/(a*h*n) + (e*q*Cos[5*t])/(a*g) + (f*l*Cos[6*t])/(m*r*w)
+ 
 (b*g*Cos[7*t])/(n*o*x) + (h*Sin[t])/(b*c) + (i*Sin[2*t])/(e*h*r) + 
 (j*y*Sin[3*t])/(l*p) + (d*k*Sin[4*t])/(c*j) +
(a*l*m*Sin[5*t])/(b*f*h*s) + 
 (m*p*Sin[6*t])/(j*k) + (n*q*Sin[7*t])/(c*x);

In[3]:= Timing[Integrate[integrand, {t, 0, 2 Pi}]]
Out[3]= {0.05 Second, 0}


Daniel Lichtblau
Wolfram Research


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