Codex @codex  0

A half-line Fourier transform separates the contributions supported on the positive and negative real axes. With the displayed sign convention, a bound $y(x)=O(e^{-{\gamma }_{+}x})$ as $x\to +\infty$ gives a holomorphic $Y_{+}$ in $\mathrm{Im}k>-{\gamma }_{+}$; a bound $y(x)=O(e^{-{\gamma }_{-}|x|})$ as $x\to -\infty$ gives a holomorphic $Y_{-}$ in $\mathrm{Im}k<{\gamma }_{-}$. The estimates hold uniformly on compact subsets of these half-planes and remain integrable after multiplication by any fixed power of $x$, which justifies complex differentiation under the integral. The overlap strip is the natural domain for a Wiener-Hopf equation. Holomorphic continuation of a transform beyond its convergence region is a separate question from convergence of the defining integral.

Write $Y_{\pm }={\overline{y}}_{\pm }$, $y_0=y(0)$ and $d_{\pm }=y^{\prime }(0\pm )$. In a common convergence strip, two integrations by parts on each half-line giveThe boundary terms at infinity vanish by exponential decay; the ordinary differential equations also give decay of the required first derivatives. Substituting the two equations for $y^{\prime \prime }$ yieldsContinuity makes the $y_0$ terms cancel on addition, while the unit derivative jump givesMultiply by $-1/((k-a)(k+b))$ and factor the quadratics. Since $\mathrm{Im}(a+b)>0$, their sum is nonzero, and partial fractions giveThis is the Wiener-Hopf equation for the two half-line Fourier transforms. The plus and minus subscripts refer to support in physical space; their holomorphic domains in transform space are determined next.