1
$\begingroup$

I have a polynomial of known degree, $f(x) = \sum_{n = 0}^N a_n x^n$ and I' d like to express it in terms of a shifted variable $x - x_ 0$, so that $f(x) = \sum_{n = 0}^N b_n (x - x_ 0)^n$.

I can do this by hand on paper, and I have a round about way of achieving the same result in Mathematica, but I' m pretty sure there has to be a standard name for the procedure I' m following when doing it manually. The round about way I currently use exploits the Taylor series, and goes like this:

Normal[Series[1 + x + x^6, {x, x0, 6}]]
Simplify[Normal[Series[1 + x + x^6, {x, x0, 6}]]] == 1 + x + x^6

Is there a more compact way of achieving this?

$\endgroup$
3
  • $\begingroup$ Check here. $\endgroup$ Mar 23 at 13:57
  • $\begingroup$ Interesting; however, that post is a close, but different, problem. I merely want a re-expression of the same function. To take their example, I want $f(x) = x^2$ to be expressed in the form $f(x) = (x+1)^2 - 2(x+1) +1$. That is, I want the polynomial in $x$ to be expressed as polynomial in $(x+1)$. $\endgroup$ Mar 23 at 15:34
  • $\begingroup$ But are they not basically inverses? So you could use that on x-1 instead of x+1, that is? $\endgroup$ Mar 23 at 15:59
1
$\begingroup$

I don't think it's more compact, but just for the sake of it, here's another way:

Collect[Expand[1 + x + x^6 /. x :> xdiff + x0], xdiff] /. xdiff :> x - x0

If you want it organized, apply HoldForm after evaluating Collect, but before replacing xdiff, which can be, for example, achieved with

HoldForm @@ {Collect[Expand[1 + x + x^6 /. x :> xdiff + x0], xdiff]} /. xdiff :> x - x0

This relies on xdiff being clear, so you may want to package it in a function which keeps your variables fresh, and which gives you the option to hold the form or not as an optional fourth argument (false by default here):

PolynomialShift[poly_, x_Symbol, x0_, hold : (True | False) : False] := 
 Module[{xdiff}, 
  If[hold, HoldForm, Identity]
    @@ {Collect[Expand[poly /. x :> xdiff + x0], xdiff]} /. xdiff :> x - x0]

(* E.g. *)

 In[1]:=  PolynomialShift[1 + x + x^6, x, x0]
Out[1]:=  1 + (x - x0)^6 + x0 + 6 (x - x0)^5 x0 + 15 (x - x0)^4 x0^2 + 
            20 (x - x0)^3 x0^3 + 15 (x - x0)^2 x0^4 + x0^6 + (x - x0) (1 + 6 x0^5)

 In[2]:=  PolynomialShift[1 + x + x^6, x, x0, True]
Out[2]:=  1 + x0 + x0^6 + (1 + 6 x0^5) (x - x0) + 15 x0^4 (x - x0)^2 + 
            20 x0^3 (x - x0)^3 + 15 x0^2 (x - x0)^4 + 6 x0 (x - x0)^5 + (x - x0)^6

If you don't care about the ordering, though,

PolynomialShift[poly_, x_Symbol, x0_] :=
  Normal[Series[poly, {x, x0, Exponent[poly, x]}]

is probably the most compact you'll get.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.