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solve/series

expressions involving general series

 Calling Sequence solve(eqn, var)

Parameters

 eqn - equation involving a series in var var - variable to be solved for

Description

 • For an equation which contains a series, solving for the series variable achieves a generalized inversion of series.
 • The result, usually a series, will be in one of the remaining indeterminates. The indeterminate which produces the simplest answer is usually chosen.
 • The global variable Order will be used to determine the order of the series result.
 • Formally, if t = solve(f(series_in_x, y), x) is a series in y with sufficient terms, then substituting t for x in $f$ will result in 0 or $\mathrm{O}\left({y}^{\mathrm{Order}}\right)$.

Examples

 > $\mathrm{Order}≔3$
 ${\mathrm{Order}}{≔}{3}$ (1)
 > $\mathrm{solve}\left(\mathrm{series}\left(x{ⅇ}^{x},x\right)=y,x\right)$
 ${y}{-}{{y}}^{{2}}{+}{\mathrm{O}}\left({{y}}^{{3}}\right)$ (2)
 > $\mathrm{solve}\left(\mathrm{series}\left({ⅇ}^{x},x\right)=a+b,x\right)$
 ${a}{-}{1}{+}{b}{-}\frac{{1}}{{2}}{}{\left({a}{-}{1}{+}{b}\right)}^{{2}}{+}{\mathrm{O}}\left({\left({a}{-}{1}{+}{b}\right)}^{{3}}\right)$ (3)