= Fourier solution of a loaded elastic half-space
{title2=$\widehat A=(1+|k|y)\widehat P e^{-|k|y}/k^2$}
For plane strain in $y>0$ under surface normal traction $-P(x)$ and zero surface shear, use the forward <Fourier transform> phase $e^{ikx}$. At $k\ne0$, biharmonicity gives $(\partial_y^2-k^2)^2\widehat A=0$, whose decaying part is $(C+Dy)e^{-|k|y}$. Boundary traction fixes $C=\widehat P/k^2$, $D=|k|C$, proving the displayed formula. Constitutive inversion and a decaying displacement mode give $2\mu\widehat v=[y+2(1-\nu)/|k|]\widehat P e^{-|k|y}$. Multiplication by $-ik$ produces an unambiguous displacement-gradient convolution kernel.
The zero mode is separate: a nonzero resultant load $\widehat P(0)$ conflicts with an ordinary fully decaying displacement normalization in two dimensions and leads to a logarithmic reference issue. Balanced loading or a distributional/reference-displacement treatment is required before extending the transformed formulas globally. Dividing by $k^2$ and assigning a finite value at $k=0$ is invalid.
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