Solving Equations with Branch Cuts - Maple Help

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Solving Equations with Branch Cuts

New functionality has been added to the solve command to account for branch cuts in the input equations and build piecewise expressions that are correct for substitution of parameters. This functionality is controlled with the new option $\mathrm{symbolic}=\mathrm{false}$. The default behavior in Maple 17 and previous versions of Maple, is the same as specifying $\mathrm{symbolic}=\mathrm{true}$.

 > $\mathrm{expr}:=a=\sqrt{a+y}+1;$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{sol1}≔\mathrm{solve}\left(\mathrm{expr},\left[y\right],\mathrm{symbolic}=\mathrm{true}\right);$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{sol2}≔\mathrm{solve}\left(\mathrm{expr},\left[y\right],\mathrm{symbolic}=\mathrm{false}\right)$
 ${\mathrm{expr}}{:=}{a}{=}\sqrt{{a}{+}{y}}{+}{1}$
 ${\mathrm{sol1}}{:=}\left[\left[{y}{=}{-}{3}{}{a}{+}{{a}}^{{2}}{+}{1}\right]\right]$
 ${\mathrm{sol2}}{:=}{{}\begin{array}{cc}\left[\left[{y}{=}{-}{3}{}{a}{+}{{a}}^{{2}}{+}{1}\right]\right]& {\mathrm{And}}{}\left({2}{}{\mathrm{argument}}{}\left({a}{-}{1}\right){\le }{\mathrm{π}}{,}{-}{\mathrm{π}}{<}{2}{}{\mathrm{argument}}{}\left({a}{-}{1}\right)\right)\\ \left[{}\right]& {\mathrm{otherwise}}\end{array}$ (1)

While both sol1 and sol2 are valid for some values of the parameter $a$:

 > $\genfrac{}{}{0}{}{\mathrm{sol1}}{\phantom{a=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{sol1}}}{a=1},\genfrac{}{}{0}{}{\mathrm{sol2}}{\phantom{a=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{sol2}}}{a=1},\mathrm{solve}\left(\genfrac{}{}{0}{}{\mathrm{expr}}{\phantom{a=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{expr}}}{a=1},\left[y\right]\right)$

Only the solution using the $\mathrm{symbolic}=\mathrm{false}$ option are valid for other values of the parameter:

 > $\genfrac{}{}{0}{}{\mathrm{sol1}}{\phantom{a=-1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{sol1}}}{a=-1},\genfrac{}{}{0}{}{\mathrm{sol2}}{\phantom{a=-1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{sol2}}}{a=-1},\mathrm{solve}\left(\genfrac{}{}{0}{}{\mathrm{expr}}{\phantom{a=-1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{expr}}}{a=-1},\left[y\right]\right)$
 $\left[\left[{y}{=}{5}\right]\right]{,}\left[{}\right]{,}\left[{}\right]$ (2)

Important Note: The default of the $\mathrm{symbolic}$ option is expected to change to in a future version of Maple. Applications that rely on the current branch cut behavior should add $\mathrm{symbolic}=\mathrm{true}$ to their calls to solve.