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Let $I_C[F]=(2\pi {)}^{-1}{\int }_C{e}^{ikx-(k^2-i\alpha k)t}F(k)dk$. The drift-diffusion half-line global relation and the half-line drift null contour identity give the formula in the secondary title. Its real-space kernel is the difference of two drifted Gaussian heat kernels, and its boundary-forcing kernel is $x(2\sqrt{\pi }{t}^{3/2}{)}^{-1}\exp [-(x+\alpha t{)}^2/(4t)]$. This kernel is an approximate identity in time as $x\downarrow 0$, which verifies the Dirichlet boundary condition.

The reflected global relation expresses the unknown value trace in terms of the prescribed derivative trace. Multiplying the half-line drift null contour identity by $(ik+\alpha )/(ik)$ removes the reflected time-dependent transform. Its pole at $k=0$ lies below the contour and is not crossed. The formula concerns the derivative $q_x(0,t)$, rather than the outward derivative $-q_x(0,t)$.

Extend $q$ by zero to $x<0$. Its ordinary Fourier transform is precisely the half-line transform $\hat{q}$, so Fourier inversion gives the real-axis term for every interior point $x>0$. At $x=0$ the zero extension can have a jump; the interior inversion is the assertion needed here.

Write $k=u+iv$. The contour is the upper branch of a hyperbola,with $u$ increasing from $-\infty$ to $\infty$. In the region above it, $\nu (k)=i\alpha -k$ has nonpositive imaginary part. Therefore $\hat{q}(\nu (k))$ is analytic there: its defining integral has factor $e^{iux-(v-\alpha )x}$ and converges for a sufficiently decaying $q$. Smoothness and integration by parts also give $\hat{q}(\nu )=O(|\nu {|}^{-1})$ away from bounded spectral sets.

For an interior spatial point $x>0$, the factor $e^{ikx}$ decays exponentially as $\mathrm{Im}k\to +\infty$. Truncate the hyperbolic contour and close it above, using the Cauchy integral theorem; the closing segments vanish by this decay and the transform estimate. No poles occur in the enclosed region. Consequently the half-line drift null contour identity isAdding any constant multiple of zero to Fourier inversion proves the proposed inversion, for every $c$. Oscillatory real-axis inversion may be understood by a Gaussian cutoff followed by its usual limit. The same contour argument allows a spectral multiplier analytic above $L$ with polynomial growth, because the exponential still controls the closing paths. We will need this slight extension for derivative boundary data.