The ComplexSolve command converts a system of complex equations in complex variables into a larger system of equations over the real numbers and solves them with real-valued equation techniques and then rewrites those solutions in terms of the original complex variables.

In general solve solves complex equations over complex variables, but notably, it does not use the full complex definition for abs and related functions like conjugate. For equations involving those functions, ComplexSolve doubles the number of variables to the real and imaginary parts of each complex variable and doubles the number of equations by taking the real and imaginary parts of each equation, and then solves the resulting system over the real numbers. Generally this is slower than calling solve directly, but may produce better solutions when the equations involve abs and conjugate.

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left\{{\mathrm{abs}\left(z\right)}^2+z\mathrm{abs}\left(z\right)+\mathrm{conjugate}\left(z\right)=0\right\}\right)$

$\left[\left[z=0\right]\,\left[z=-\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}\right]\,\left[z=-\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}\right]\right]$

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left\{{\mathrm{abs}\left(z\right)}^2+z\mathrm{abs}\left(z\right)+\frac{{\mathrm{abs}\left(z\right)}^2}{z}=0\right\}\right)$

$\left[\left[z=0\right]\,\left[z=-\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}\right]\,\left[z=-\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}\right]\right]$

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left\{\mathrm{abs}\left(z\right)=1\right\}\right)$

$\left[\left[z=\sqrt{-{\mathrm{\_t1}}^2+1}+\mathrm{I}\mathrm{\_t1}\&comma;0<\sqrt{-{\mathrm{\_t1}}^2+1}\right]\&comma;\left[z=\mathrm{-I}\right]\&comma;\left[z=\mathrm{I}\right]\&comma;\left[z=-\sqrt{-{\mathrm{\_t2}}^2+1}+\mathrm{I}\mathrm{\_t2}\&comma;-\sqrt{-{\mathrm{\_t2}}^2+1}<0\right]\right]$

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left\{\mathrm{abs}\left(z\right)=\mathrm{I}\right\}\right)$

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left[z^8-{\mathrm{abs}\left(z\right)}^7\&comma;z-\mathrm{conjugate}\left(z\right)\mathrm{I}\right]\right)$

$\left[\left[z=0\right]\&comma;\left[z=-\frac{\sqrt{2}}{2}-\frac{\mathrm{I}\sqrt{2}}{2}\right]\&comma;\left[z=\frac{\sqrt{2}}{2}+\frac{\mathrm{I}\sqrt{2}}{2}\right]\right]$

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left[z^8-{\mathrm{abs}\left(z\right)}^7\right]\right)$

$\left[\left[z=0\right]\&comma;\left[z=\mathrm{I}\right]\&comma;\left[z=\mathrm{-I}\right]\&comma;\left[z=1\right]\&comma;\left[z=\mathrm{-1}\right]\&comma;\left[z=\frac{\sqrt{2}}{2}-\frac{\mathrm{I}\sqrt{2}}{2}\right]\&comma;\left[z=-\frac{\sqrt{2}}{2}+\frac{\mathrm{I}\sqrt{2}}{2}\right]\&comma;\left[z=-\frac{\sqrt{2}}{2}-\frac{\mathrm{I}\sqrt{2}}{2}\right]\&comma;\left[z=\frac{\sqrt{2}}{2}+\frac{\mathrm{I}\sqrt{2}}{2}\right]\right]$

$\mathrm{SolveTools}:-\mathrm{ComplexSolve}\left(\left[z^4+{\mathrm{conjugate}\left(z\right)}^4\right]\right)$

$\left[\left[z=\left(1+\sqrt{2}\right)\mathrm{\_t3}+\mathrm{I}\mathrm{\_t3}\&comma;0<\left(1+\sqrt{2}\right)\mathrm{\_t3}\right]\&comma;\left[z=0\right]\&comma;\left[z=\left(1+\sqrt{2}\right)\mathrm{\_t3}+\mathrm{I}\mathrm{\_t3}\&comma;\left(1+\sqrt{2}\right)\mathrm{\_t3}<0\right]\&comma;\left[z=\left(1-\sqrt{2}\right)\mathrm{\_t4}+\mathrm{I}\mathrm{\_t4}\&comma;0<\left(1-\sqrt{2}\right)\mathrm{\_t4}\right]\&comma;\left[z=\left(1-\sqrt{2}\right)\mathrm{\_t4}+\mathrm{I}\mathrm{\_t4}\&comma;\left(1-\sqrt{2}\right)\mathrm{\_t4}<0\right]\&comma;\left[z=\left(\sqrt{2}-1\right)\mathrm{\_t5}+\mathrm{I}\mathrm{\_t5}\&comma;0<\left(\sqrt{2}-1\right)\mathrm{\_t5}\right]\&comma;\left[z=\left(\sqrt{2}-1\right)\mathrm{\_t5}+\mathrm{I}\mathrm{\_t5}\&comma;\left(\sqrt{2}-1\right)\mathrm{\_t5}<0\right]\&comma;\left[z=\left(-1-\sqrt{2}\right)\mathrm{\_t6}+\mathrm{I}\mathrm{\_t6}\&comma;0<\left(-1-\sqrt{2}\right)\mathrm{\_t6}\right]\&comma;\left[z=\left(-1-\sqrt{2}\right)\mathrm{\_t6}+\mathrm{I}\mathrm{\_t6}\&comma;\left(-1-\sqrt{2}\right)\mathrm{\_t6}<0\right]\right]$

The SolveTools:-ComplexSolve command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.