= Solution
Use the <Unscaled Papkovich–Neuber representation> with $e_k=e^{ikx-ky}$ and harmonic potentials
$$
\boxed{\Phi=(0,iUe_k),\qquad\chi=-\frac{iU}{k}e_k.}
$$
To see how these coefficients arise, start with $\Phi=(0,Ae_k)$ and $\chi=Be_k$. The representation gives
$$
u=ik(Ay+B)e_k,\qquad v=(-A-kB-kAy)e_k.
$$
At $y=0$, the prescribed tangential velocity requires $ikB=U$, while zero normal velocity requires $A=-kB=iU$. Hence the \b[decaying half-space flow] is
$$
\boxed{u=U(1-ky)e_k,\qquad v=-ikUy e_k,\qquad p=-2i\mu kUe_k.}
$$
The pressure follows from $2\mu\partial_y\Phi_y$. These harmonic potentials verify the <Stokes equation> and <incompressibility>; the polynomial factors do not prevent decay as $y\to\infty$.
For the <Newtonian fluid stress tensor>, $\sigma_{xy}=\mu(u_y+v_x)$ and $\sigma_{yy}=-p+2\mu v_y$. On the boundary $u_y=-2kUe^{ikx}$, $v_x=0$ and $v_y=-ikUe^{ikx}$. Therefore
$$
\boxed{\sigma_{xy}(x,0)=-2\mu kUe^{ikx},\qquad\sigma_{yy}(x,0)=0.}
$$
Take real parts for a physical sinusoidal flow. Together with the normal-velocity result supplied in the question, these formulas give the <Fourier traction map for a viscous half-space>; for signed wave numbers its resistance coefficient is $2\mu|k|$.
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