= Half-line Fourier transform
{title2=$Y_+(k)=\int_0^\infty y(x)e^{ikx}\,dx,\quad Y_-(k)=\int_{-\infty}^0y(x)e^{ikx}\,dx$}
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 $\operatorname{Im}k>-\gamma_+$; a bound $y(x)=O(e^{-\gamma_-|x|})$ as $x\to-\infty$ gives a <holomorphic> $Y_-$ in $\operatorname{Im}k<\gamma_-$. 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>.
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