= Solution
Since $u$ and $v$ are <harmonic conjugates>, $f=u+iv$ is a <holomorphic function> on $D$, with $u_x=v_y$ and $u_y=-v_x$. The entire function $w\mapsto e^{iw^2}$ can be composed with $f$. Its value is
$$
e^{if^2}=e^{i(u^2-v^2+2iuv)}=e^{-2uv}\bigl[\cos(u^2-v^2)+i\sin(u^2-v^2)\bigr]=U+iV.
$$
By the complex <chain rule> the composition is holomorphic on $D$, so its real and imaginary parts obey the <Cauchy-Riemann equations>. They are harmonic as well: differentiating those equations gives $U_{xx}+U_{yy}=V_{yx}-V_{xy}=0$ and likewise $V_{xx}+V_{yy}=0$. Hence \b[$U,V$ are a pair of harmonic conjugates on $D$]. No simple-connectivity assumption is needed because the conjugate pair is already supplied globally. The PDF has $e^{-2uv}$ in both expressions; the differing TeX exponents are transcription defects.
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