The command Substitute(x = s, p) computes the result of substituting the expression s for the variable x in the power series p. The expression s can be a polynomial or a power series generated by this package.

The calling sequence Substitute(ls, p), where ls is a list or set of equations of the form x = s above, applies all of the given substitutions simultaneously. More precisely, if ls contains the equations x[i] = s[i] where i runs from 1 to n, then it returns an answer equivalent to what would be obtained from the following process:

For each variable v[i,j] occurring in s[i], replace it with a variable u[i,j] that does not occur anywhere else (it should also not occur as u[i,j] in previous such replacements). Call the result t[i].

Perform the substitutions x[i] = t[i] in any order. Call the result q.

Replace each u[i, j] in q with the original v[i, j].

An error is signaled if the same variable x occurs more than once as a left hand side in ls.

If s is not a unit, that is, if its constant coefficient is zero, then the result of the substitution is easily defined by just substituting one power series into another; one need consider only finitely many terms to find any homogeneous component of the result. However, if s is a unit, then the result of the substitution is a priori not well-defined. In these cases, Maple uses the analytic expression specified for p in a computation that gives a result that is correct in a formal sense. If the analytic expression for p is not specified, Maple signals an error. If a list or set of substitutions is specified and some of the right hand sides s are unit power series, then this computation only works if p and at least all but one of these unit power series have their analytic expression specified.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

$\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right)\:$

$\mathrm{p1}≔x^2+yz+z$

$\mathrm{p1}≔x^2+yz+z$

$\mathrm{p2}≔y^2+zx+x$

$\mathrm{p2}≔zx+y^2+x$

$\mathrm{p3}≔z^2+xy+y$

$\mathrm{p3}≔xy+z^2+y$

$\mathrm{ps1},\mathrm{ps2},\mathrm{ps3}≔\mathrm{seq}\left(\mathrm{PowerSeries}\left(p\right)\,p\mathbf{in}\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\right)$

$\mathrm{ps1},\mathrm{ps2},\mathrm{ps3}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}z+x^2+yz\right],\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}x+zx+y^2\right],\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}y+xy+z^2\right]$

$\mathrm{result}≔\mathrm{Substitute}\left(\left\{x=\mathrm{ps1}\,y=\mathrm{ps2}\right\}\,\mathrm{ps3}\right)$

$\mathrm{result}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\left(x^2+yz+z\right)\left(zx+y^2+x\right)+z^2+zx+y^2+x\mathrm{:}0+\mathrm{\dots }\right]$

$\mathrm{ApproximatelyEqual}\left(\mathrm{result}\,\mathrm{Substitute}\left(\left\{x=\mathrm{p1}\,y=\mathrm{p2}\right\}\,\mathrm{ps3}\right)\,4\right)$

$\mathrm{true}$

$\mathrm{expand}\left(\mathrm{Truncate}\left(\mathrm{result}\,4\right)-\mathrm{subs}\left(\left\{x=\mathrm{p1}\,y=\mathrm{p2}\right\}\,\mathrm{p3}\right)\right)$

$0$

$\mathrm{q1},\mathrm{q2},\mathrm{q3}≔\mathrm{seq}\left(p+1\,p\mathbf{in}\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\right)$

$\mathrm{q1},\mathrm{q2},\mathrm{q3}≔x^2+yz+z+1,zx+y^2+x+1,xy+z^2+y+1$

$\mathrm{qs1},\mathrm{qs2},\mathrm{qs3}≔\mathrm{seq}\left(\mathrm{PowerSeries}\left(q\right)\,q\mathbf{in}\left[\mathrm{q1}\,\mathrm{q2}\,\mathrm{q3}\right]\right)$

$\mathrm{qs1},\mathrm{qs2},\mathrm{qs3}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+z+x^2+yz\right],\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+x+zx+y^2\right],\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+y+xy+z^2\right]$

$\mathrm{qresult}≔\mathrm{Substitute}\left(\left\{x=\mathrm{qs1}\,y=\mathrm{qs2}\right\}\,\mathrm{qs3}\right)$

$\mathrm{qresult}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\left(x^2+yz+z+1\right)\left(zx+\mathrm{\dots }+\mathrm{\dots }\right)+z^2+zx+y^2+x+2\mathrm{:}3+\mathrm{\dots }\right]$

$\mathrm{ApproximatelyEqual}\left(\mathrm{qresult}\,\mathrm{Substitute}\left(\left\{x=\mathrm{q1}\,y=\mathrm{q2}\right\}\,\mathrm{qs3}\right)\,4\right)$

$\mathrm{true}$

$\mathrm{expand}\left(\mathrm{Truncate}\left(\mathrm{qresult}\,4\right)-\mathrm{subs}\left(\left\{x=\mathrm{q1}\,y=\mathrm{q2}\right\}\,\mathrm{q3}\right)\right)$

$0$

$\mathrm{exp\_x}≔\mathrm{PowerSeries}\left(d\mapsto \frac{x^d}{d!}\,\mathrm{analytic}=\exp \left(x\right)\right)$

$\mathrm{exp\_x}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^x\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{result}≔\mathrm{Substitute}\left(x=\mathrm{exp\_x}\,\mathrm{exp\_x}\right)$

$\mathrm{result}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^{{\ⅇ}^x}\mathrm{:}\ⅇ+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(\mathrm{result}\,4\right)$

$\ⅇ+\ⅇx+\ⅇx^2+\frac{5\ⅇx^3}{6}+\frac{5\ⅇx^4}{8}$

$\mathrm{series}\left(\exp \left(\exp \left(x\right)\right)\,x=0\,5\right)$

$\ⅇ+\ⅇx+\ⅇx^2+\frac{5}{6}\ⅇx^3+\frac{5}{8}\ⅇx^4+O\left(x^5\right)$

$\mathrm{exp\_y\_no\_a}≔\mathrm{PowerSeries}\left(d\mapsto \frac{y^d}{d!}\,\mathrm{variables}=\left\{y\right\}\right)$

$\mathrm{exp\_y\_no\_a}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+\mathrm{\dots }\right]$

$\mathrm{base}≔\mathrm{PowerSeries}\left(\mathrm{ex}+\mathrm{ey}\right)$

$\mathrm{base}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}\mathrm{ex}+\mathrm{ey}\right]$

$\mathrm{step1}≔\mathrm{Substitute}\left(\mathrm{ey}=\mathrm{exp\_y\_no\_a}\,\mathrm{base}\right)$

$\mathrm{step1}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+\mathrm{\dots }\right]$

$\mathrm{step2}≔\mathrm{Substitute}\left(\mathrm{ex}=\mathrm{exp\_x}\,\mathrm{step1}\right)$

Error, (in MultivariatePowerSeries:-Substitute) substituting a unit power series into a power series with undefined analytic expression

$\mathrm{alt\_step1}≔\mathrm{Substitute}\left(\mathrm{ex}=\mathrm{exp\_x}\,\mathrm{base}\right)$

$\mathrm{alt\_step1}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^x+\mathrm{ey}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{alt\_step2}≔\mathrm{Substitute}\left(\mathrm{ey}=\mathrm{exp\_y\_no\_a}\,\mathrm{alt\_step1}\right)$

$\mathrm{alt\_step2}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}2+\mathrm{\dots }\right]$

$\mathrm{result}≔\mathrm{Substitute}\left(\left\{\mathrm{ex}=\mathrm{exp\_x}\,\mathrm{ey}=\mathrm{exp\_y\_no\_a}\right\}\,\mathrm{base}\right)$

$\mathrm{result}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}2+\mathrm{\dots }\right]$

$\mathrm{ApproximatelyEqual}\left(\mathrm{result}\,\mathrm{alt\_step2}\,4\right)$

$\mathrm{true}$

The MultivariatePowerSeries[Substitute] command was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.