Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 69 / 2

Use the Wirtinger derivatives ${\partial }_z=({\partial }_x-i{\partial }_y)/2$ and ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$, and area measure $dA=dxdy=(i/2)dz\wedge d\bar{z}$. The supplied boundary-integral identity is the planar Generalized Stokes theorem; the usual Poincare lemma is a different local exactness result.

Let $f\in C^1(\overline{D})$ and $z\in D$. Remove a disk of radius $ϵ$ around $z$, and apply Generalized Stokes theorem to the one-form $f(\zeta )d\zeta /(\zeta -z)$ on the punctured domain. Away from the puncture,The outer boundary is counterclockwise and the small inner circle clockwise. Its counterclockwise integral tends to $2\pi if(z)$. Passing to the limit gives the Cauchy-Pompeiu formulaThe weak $1/|\zeta -z|$ singularity is locally integrable. If the boundary term vanishes on expanding $D$ to the whole plane, the formula becomes $f(z)={\pi }^{-1}{\int }_{\mathbb{C}}{f}_{\bar{\zeta }}(\zeta )/(z-\zeta )dA(\zeta )$. In particular it yields the distributional normalization ${\partial }_{\bar{z}}[1/(\pi z)]={\delta }_0$. For holomorphic functions the area term vanishes and one recovers the Cauchy integral formula.

Work first with $q$ in the Schwartz space, so that all Fourier manipulations and spectral contour integrals are justified; weaker classes follow by the usual density or distribution arguments. DefineThe phase is purely imaginary. Since ${\partial }_{\bar{z}}E=kE$, setting $F=E\Phi$ reduces the spectral equation to ${\partial }_{\bar{z}}\Phi =E^{-1}q$. The whole-plane Cauchy-Pompeiu formula therefore constructs the solution decaying spatially at infinity:The freedom to add $E$ times an entire function is removed by this decay condition. For every fixed $k$, the integral is a spatial Cauchy-Green operator applied to a modulated source.

Now differentiate in the conjugate spectral parameter. The two exponential derivatives produce $\zeta -z$, canceling the Cauchy denominator, soThis is the spectral dbar equation: its right-hand side is the forward transform of $q$ multiplied by a known plane wave. Apply the whole-plane Cauchy-Pompeiu formula again, now in $k$:The spatial spectral equation also gives $F(z,k)=-q(z)/k+o(k^{-1})$ as $|k|\to \infty$. One way to justify this is to integrate by parts in the first Cauchy integral: $F=-q/k+T_k(q_{\bar{z}})/k$, where the modulated Cauchy integral $T_k(q_{\bar{z}})$ tends to zero by the Riemann-Lebesgue lemma. Comparing the $1/k$ coefficient of the spectral contour integral therefore givesThis derives the transform pair from two uses of the Cauchy-Pompeiu formula, not from an assumed inversion formula.

To identify the usual normalization, write $z=x+iy$ and $k=k_1+i{k}_2$. Then $\bar{k}z-k\bar{z}=2i(k_{1}y-k_{2}x)$. Set ${\xi }_1=2k_2$, ${\xi }_2=-2k_1$, so $d{\xi }_{1}d{\xi }_2=4dA(k)$. The result is exactly the two-dimensional Fourier transform pairThe factor four in the real-frequency change of variables is essential.

Assume for this spectral construction that the source and known attenuation are sufficiently smooth and decaying, for instance compactly supported, and that $\mu \ge 0$ for the physical interpretation. The following steps identify both the forward attenuated Radon transform and the route to its inversion.

First write the complex transport operator asFor $k=e^{i\theta }$ this is the real directional derivative ${\mathbf{d}}_{\theta }\cdot \nabla$, with ${\mathbf{d}}_{\theta }=(\cos \theta ,\sin \theta )$. Put ${\mathbf{n}}_{\theta }=(-\sin \theta ,\cos \theta )$ and $x=p{\mathbf{n}}_{\theta }+s{\mathbf{d}}_{\theta }$. The spectral equation on the unit circle becomesIts minus sign fixes the appropriate endpoint condition: use $F(p,+\infty )=0$. The integrating factor givesThus the measured quantity at the opposite end of the line isThe exponent is the attenuation accumulated between the source point and the detector at the negative end. The common convention with detector at the positive end is the same transform after reversing the direction, ${\mathcal{R}}_{\mu }^{+}f(p,\theta )={\mathcal{R}}_{\mu }^{-}f(-p,\theta +\pi )$. Using an incoming zero condition at the negative end while retaining the printed minus sign would instead give a growing integrating factor, not physical attenuation.

Next, for $|k|\ne 1$, the operator $L_k$ is a complex elliptic first-order operator. Its decaying whole-plane Green function isThis is obtained by a real-linear change of variables in the Cauchy-Green operator; its change of orientation explains the sign. Write $G_{k}a$ for convolution with this kernel. Solve $L_{k}h=\mu$, and set $F=e^h\Psi$. This removes the attenuation:The normalized spectral solution is analytic separately inside and outside the unit circle. As $k$ approaches that circle, the Green function's characteristic singularity produces two limiting values. Their relation is computed from the weighted line integrals above together with transverse Hilbert transforms; the known attenuation determines the integrating factor weights. The two spectral limits are not individually just the incoming and outgoing real characteristic solutions: the singular-kernel prescription matters.

Finally formulate the resulting additive Riemann-Hilbert problem on the unit circle. Its jump is determined by the measured attenuated Radon transform and known $\mu$. With the unit circle oriented counterclockwise and jump $J=F_{\mathrm{inside}}-F_{\mathrm{outside}}$, a Cauchy integral formula reconstructs the normalized spectral solution:The normalization at zero supplies the compatibility condition $\int J/\zeta d\zeta =0$. Recover $f$ from $L_{k}F-\mu F$, or from its small-$k$ coefficient: if $F=k{F}_1+O(k^2)$, then $f={\partial }_{\bar{z}}{F}_1$. Equivalently the large-$k$ coefficient gives $f={\partial }_z{F}_{-1}$ when $F=k^{-1}{F}_{-1}+O(k^{-2})$. This spectral reconstruction of an attenuated Radon transform is the analogue of recovering $q$ from a coefficient in (ii). Known attenuation and full directed line data are inputs; one does not determine an arbitrary unknown attenuation and source simultaneously from this argument.