= Solution
Set $A=x^2-y^2$ and $B=2xy$. Direct <differentiation> gives
$$
u_{xx}=e^{-B}[(4y^2-4x^2)\sin A+(2-8xy)\cos A],\qquad
u_{yy}=e^{-B}[(4x^2-4y^2)\sin A+(8xy-2)\cos A].
$$
Their sum vanishes, proving that $u$ is a <harmonic function>. More generally the <Cauchy-Riemann equations> give $u_{xx}+u_{yy}=v_{yx}-v_{xy}=0$ and $v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0$; the <mixed partial derivatives> commute because a <holomorphic function> is smooth. Since $e^{iz^2}=e^{-B}(\cos A+i\sin A)$, a <harmonic conjugate> and the resulting <holomorphic function> are
$$
\boxed{v=-e^{-2xy}\cos(x^2-y^2)+C,\qquad f(z)=-ie^{iz^2}+iC,\quad C\in\mathbb R.}
$$
These expressions also verify the <Cauchy-Riemann equations> directly.
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