Let $z=x+iy$ and $v=2xy$, show that $v=Im[z^2]$ and find a harmonic conjugate of $v$ on domain $D$. Also find the largest domain $D$ on which $v$ is harmonic.
2026-04-12 11:33:07.1775993587
Imaginary complex numbers
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$z^2=(x+iy)^2=x^2+2xiy+i^2y^2=...$ continue then conclude with its imaginary part.
The holomorphic function would be $z\mapsto z^2$, it is defined on all $\Bbb C$. The harmonic conjugate would be then $u:=z\mapsto Re[z^2]$. Verify that these satisfy the Cauchy-Riemann equations.