appendix.tex

\chapter*{Appendix}
\markboth{APPENDIX}{}
\addcontentsline{toc}{chapter}{Appendix}
\label{chap:app}

In this chapter, we will provide the missing calculations for the proofs of Lemma~\ref{lem:HelmholtzLaplaceDifference} and Theorem~\ref{thm:differenceFundamentalSolutionStokes}.
Therefore, we will build on the notations which were already established at the beginning of Chapter~\ref{chap:2}.

We first collect expressions for the derivatives of the fundamental solutions to the scalar Helmholtz equation and the Laplace equation in $d= 2$.
For the fundamental solution to the Laplace equation $G(x; 0) = -\frac{1}{2\pi} \log(|x|)$, the partial derivatives read:
\begin{align*}
  \partial_\gamma G(x; 0) &= -\frac{1}{2\pi} \, \frac{x_\gamma}{|x|^2}, \\[0.5em]
  %
  \partial_\alpha \partial_\gamma G(x; 0) &= -\frac{1}{2\pi}\, \frac{\delta_{\alpha\gamma}}{|x|^2} + \frac{1}{\pi} \, \frac{x_\alpha x_\gamma}{|x|^4}, \\[0.5em]
  %
  \partial_\beta \partial_\alpha \partial_\gamma G(x; 0)
  &= \frac{1}{\pi} \, \frac{\delta_{\beta \gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^4} - \frac{4}{\pi}\, \frac{x_\alpha x_\beta x_\gamma}{|x|^6}.
\end{align*}
The fundamental solution for the scalar Helmholtz equation is given via 
\begin{align*}
  G(x; \lambda) = \frac{\ii}{4}\, H_0^{(1)}(k|x|),
\end{align*}
where $H_0^{(1)}(z)$ is the Hankel function of the first kind, see Section \ref{sec:hankel}.
Complex derivatives will be denoted by $\frac{\d{}}{\d z}$.
We calculate using the chain rule and the product rule:
\begin{align*}
  %%
  \partial_\gamma G(x; \lambda) &= \frac{\ii}{4}\, k\;  \frac{x_\gamma}{|x|}\, \frac{\d{}}{\d z} H_0^{(1)}(k |x|)\,, \\[0.5em]
  %%
  \partial_\alpha \partial_\gamma G(x; \lambda) 
  %%
  &=  \frac{\ii}{4}\,  k \,\bigg(\frac{\delta_{\alpha\gamma}}{|x|} - \frac{x_\alpha x_\gamma}{|x|^3} \bigg)\, \frac{\d{}}{\d z} H_0^{(1)}(k |x|) + \frac{\ii}{4} \, k^2 \;  \frac{x_\alpha x_\gamma}{|x|^2} \, \frac{\d{}^2}{\d z^2} H_0^{(1)}(k |x|)\,, \\[0.5em]
  %%
  \partial_\beta \partial_\alpha \partial_\gamma G(x; \lambda)
  %%
  &= \frac{\ii}{4} \, k^3 \; \frac{x_\alpha x_\beta x_\gamma}{|x|^3} \, \frac{\d{}^3}{\d z^3} H_0^{(1)}(k |x|)\\[0.5em]
  %
  &\quad + \frac{\ii}{4} \, k^2 \,\bigg(\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \bigg)\, \frac{\d{}^2}{\d z^2} H_0^{(1)}(k |x|) \\[0.5em]
  %
  &\quad+ \frac{\ii}{4} \, k \, \bigg( 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \bigg)\, \frac{\d{}}{\d z} H_0^{(1)}(k |x|) \,.
\end{align*}

We will now present the derivatives of the Hankel function $H_0^{(1)}(z)$ which appeared in the previous calculations by calculating the complex derivatives of the series expansion of $H_0^{(1)}(z)$ which may be found in Lebedev \cite[Sec.\@~5.6]{lebedev}. 
The expansions below are a consequence of the series expansions of the Bessel functions of the first and second kind $J_0$ and $Y_0$, respectively.
In the following, $\psi$ will denote the \emph{Digamma function}. 
The aforementioned expansions read:
\begin{align*}
  %
  \frac{\pi}{2 \ii} \, H_0^{(1)}(z)
  %
  &= \frac{\pi}{2\ii} \, J_0(z) + \frac{\pi}{2}\, Y_0(z) \\
  &= \mathlarger{\sum}_{l = 0}^\infty \, \frac{(-1)^l}{(l!)^2 \, 4^l} \,  z^{2l} \Big( -\frac{\ii \pi}{2} - \log(2) - \psi(l + 1) \Big) 
  +  \mathlarger{\sum}_{l = 0}^\infty \, \frac{(-1)^l}{(l!)^2 \, 4^l} \,  z^{2l} \log(z) \\
  %&= \frac{2 \ii}{\pi} \, \mathlarger{\sum}_{l = 0}^\infty \, \frac{(-1)^l}{(l!)^2 \, 4^l} \,  z^{2l} \bigg( -\frac{\ii \pi}{2} - \log(2) - \psi(l + 1) \bigg) + \frac{2 \ii}{\pi} \, \mathlarger{\sum}_{l = 0}^\infty \, \frac{(-1)^l}{(l!)^2 \, 4^l} \,  z^{2l} \log(z) \\
  &= \mathlarger{\sum}_{l = 0}^\infty \, a_l \,  z^{2l} \, C_l 
   + \mathlarger{\sum}_{l = 0}^\infty \, a_l  \,z^{2l} \log(z)\,, \\[1.0em]
  %
  \frac{\pi}{2\ii} \, \frac{\d{}}{\d z} H_0^{(1)}(z)
  %
  &= \mathlarger{\sum}_{l = 1}^\infty \, a_l \,  (2l) \, z^{2l - 1} \, C_l 
  + \mathlarger{\sum}_{l = 1}^\infty \, a_l \, (2l)  \, z^{2l - 1} \log(z) \, 
  + \mathlarger{\sum}_{l = 0}^\infty \, a_l \, z^{2l - 1} \\
  &= \mathlarger{\sum}_{l = 1}^\infty \, b_l \, z^{2l - 1} \, C_l 
  + \mathlarger{\sum}_{l = 1}^\infty \, b_l \, z^{2l - 1} \log(z) \, 
  +  \mathlarger{\sum}_{l = 0}^\infty \, a_l \, z^{2l - 1}, \\[1.0em]
  %
  \frac{\pi}{2\ii} \, \frac{\d{}^2}{\d z^2} H_0^{(1)}(z)
  %
  &= \mathlarger{\sum}_{l = 1}^\infty \, b_l \, (2l - 1) \, z^{2l - 2} \, C_l 
  + \mathlarger{\sum}_{l = 1}^\infty \, b_l \, (2l - 1)\, z^{2l - 2} \log(z) 
  + \mathlarger{\sum}_{l = 1}^\infty \, b_l \, z^{2l - 2}\\
  &\quad + \mathlarger{\sum}_{l = 0}^\infty \, a_l \, (2l - 1) \, z^{2l - 2} \\
  &=   \mathlarger{\sum}_{l = 1}^\infty c_l \, z^{2l - 2} \, C_l 
  +  \!\mathlarger{\sum}_{l = 1}^\infty c_l \, z^{2l - 2} \log(z) 
  +  \!\mathlarger{\sum}_{l = 1}^\infty b_l \, z^{2l - 2} 
  +  \!\mathlarger{\sum}_{l = 0}^\infty a_l \, (2l - 1) \, z^{2l - 2}, \\[1.0em]
  %
  \frac{\pi}{2\ii} \, \frac{\d{}^3}{\d z^3} H_0^{(1)}(z)
  %
  &= \sum_{l = 2}^\infty c_l \, (2l - 2) \, z^{2l - 3} \, C_l 
  +  \mathlarger{\sum}_{l = 2}^\infty \, c_l \, (2l - 2) \, z^{2l - 3} \log(z) 
  +  \mathlarger{\sum}_{l = 1}^\infty \, c_l \, z^{2l - 3} \\
 &\quad  
  + \mathlarger{\sum}_{l = 2}^\infty \, b_l \, (2l - 2) \, z^{2l - 3} 
  + \mathlarger{\sum}_{l = 0}^\infty \, a_l \, (2l - 1) \, (2l - 2) \, z^{2l - 3} \\
  %%
  &= \mathlarger{\sum}_{l = 2}^\infty \, d_l \, z^{2l - 3} \, C_l 
  + \mathlarger{\sum}_{l = 2}^\infty \, d_l \, z^{2l - 3} \log(z) 
  + \mathlarger{\sum}_{l = 1}^\infty \, c_l \, z^{2l - 3} \\
 &\quad  + \mathlarger{\sum}_{l = 2}^\infty \, b_l \, (2l - 2) \, z^{2l - 3} 
  + \mathlarger{\sum}_{l = 0}^\infty \, a_l \, (2l - 1)\, (2l - 2) \, z^{2l - 3} ,
\end{align*}
where, in order to increase readability, we introduced the following coefficients:
\begin{align}
  \begin{alignedat}{1}
  C_l &\coloneqq -\frac{\ii \pi}{2} - \log(2) - \psi(l + 1), \\
  a_l &\coloneqq \frac{(-1)^l}{(l!)^2 \, 4^l}, \quad
  b_l \coloneqq a_l \cdot  2l, \quad
  c_l \coloneqq b_l \cdot (2l - 1) \quad\text{and}\quad
  d_l \coloneqq c_l \cdot (2l - 2).
  \end{alignedat}\label{defnConst}\tag{D1}
\end{align}

\section*{A.1\quad Proof of Lemma \ref{lem:HelmholtzLaplaceDifference} for $d = 2$}
\markboth{APPENDIX}{A.1.\quad PROOF OF LEMMA \ref*{lem:HelmholtzLaplaceDifference} FOR $d = 2$}
\addcontentsline{toc}{section}{A.1\quad Proof of Lemma \ref*{lem:HelmholtzLaplaceDifference} for $d = 2$}
\label{sec:A1}

For $\lambda \in \Sigma_\theta$, $\theta \in (0, \pi/2)$, we need to show that for $|\lambda| |x|^2 \leq (1/2)$ the estimate
\begin{align*}
  \Big|\, \partial_\beta \partial_\alpha \partial_\gamma \big\{ G(x; \lambda) - G(x; 0) \big\} \, \Big|
  \leq C\, |\lambda| |x|^{-1}
\end{align*}
holds with a constant $C > 0$ that only depends on $\theta$.
The strategy will be to first estimate all absolute values of the resulting terms individually and to extract those which cannot be estimated in this way.
We will call the terms whose absolute value that cannot be bounded individually \emph{problematic}.
In the second step, we will show that all problematic terms cancel when added together which shows that the claimed estimate holds.

In order to make the next calculations better to digest, we decompose the third derivative of $G(\,\cdot\, ; \lambda)$ as follows:
\begin{align*}
  \partial_\beta \partial_\alpha \partial_\gamma G(x; \lambda)
  = A_3 + A_2 + A_1,
\end{align*}
where each $A_i$, $i = 1,\dots,3$, corresponds to the term involving the $i$th derivative of $H_0^{(1)}$.
Let us start with $A_3$. 
We have
\begin{align*}
  A_3 = - \frac{1}{2 \pi } k^3 \, \frac{x_\alpha x_\beta x_\gamma}{|x|^3}  \,
  %%
  &\cdot\,\bigg\{
     \; \mathlarger{\sum}_{l = 2}^\infty \; d_l \, (k|x|)^{2l - 3} \, C_l +  \mathlarger{\sum}_{l = 2}^\infty \; d_l \, (k|x|)^{2l - 3} \log(k|x|)  \\
  &\quad\,  +  \mathlarger{\sum}_{l = 1}^\infty \; c_l \, (k|x|)^{2l - 3} +  \mathlarger{\sum}_{l = 2}^\infty \; b_l\, (2l - 2) \, (k|x|)^{2l - 3} \\
  &\quad\, +  \mathlarger{\sum}_{l = 0}^\infty \; a_l \, (2l - 1)\, (2l - 2) \, (k|x|)^{2l - 3} \;
  \bigg\}.
  %%
\end{align*}
First, note that
\begin{align*}
  \Big|\, \frac{1}{2\pi} k^3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^3} \,\cdot\; c_1\, (k|x|)^{2 \cdot 1 -3} \,\Big| 
  \leq C \, |\lambda| |x|^{-1},
\end{align*}
with a constant $C > 0$.
This shows that the first term of the third sum in $A_3$ is not problematic.
For the rest of $A_3$ we use the fact $|\lambda| |x|^2 \leq (1/2)$ to trade one $|k| = \sqrt{|\lambda|}$ in the prefactor for a constant times $|x|^{-1}$ and show that the prefactor behaves as
\begin{align*}
  \Big|\, k^3 \, \frac{x_\alpha x_\beta x_\gamma}{|x|^3}\, \Big| 
  \leq C\, |\lambda| |x|^{-1}, 
\end{align*}
with a constant $C > 0$.
Therefore, the only problematic term in $A_3$ is the first element of the last sum
\begin{align}
  \label{eq:P1}
  %\tag{P1}- \frac{1}{2 \pi } k^3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^3}  \; \mathlarger{\sum}_{l = 0}^{0} \; a_l \, (2l - 1)\, (2l - 2) \, (k|x|)^{2l - 3} .
  \tag{P1}- \frac{1}{2 \pi } k^3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^3} \, \cdot\;  a_0 \, (2 \cdot 0 - 1)\, (2 \cdot 0 - 2) \, (k|x|)^{2 \cdot 0 - 3} .
\end{align}
For $A_2$, we calculate
\begin{align*}
  A_2 &= - \frac{1}{2\pi} k^2\, \bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) \\ 
  &\qquad \cdot 
  \bigg\{
    \; \mathlarger{\sum}_{l = 1}^\infty \; c_l \, (k|x|)^{2l - 2} \, C_l 
  + \; \mathlarger{\sum}_{l = 1}^\infty \; c_l \, (k|x|)^{2l - 2} \log(k|x|) \\
  &\qquad\quad\,+ \; \mathlarger{\sum}_{l = 1}^\infty \; b_l \, (k|x|)^{2l - 2} 
  + \; \mathlarger{\sum}_{l = 0}^\infty \; a_l\, (2l - 1) \, (k|x|)^{2l - 2} 
  \bigg\}.
\end{align*}
As the prefactor already behaves like $|\lambda| |x|^{-1}$, we identify the following to terms as being problematic:
\begin{align}
  \label{eq:P2}
  \tag{P2}
  \begin{alignedat}{1}
%  & - \frac{1}{2\pi} k^2\, \bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
%   \, \cdot \mathlarger{\sum}_{l = 1}^1 \; c_l \, (k|x|)^{2l - 2} \log(k|x|) \\
%   &- \frac{1}{2\pi} k^2 \,\bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
%  \, \cdot \mathlarger{\sum}_{l = 0}^0 \; a_l\, (2l - 1) \, (k|x|)^{2l - 2} .
    %
  & - \frac{1}{2\pi} k^2\, \bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
    \, \cdot\;  c_1 \, (k|x|)^{2 \cdot 1 - 2} \log(k|x|) \;\;\;\text{and}\\[0.5em]
   &- \frac{1}{2\pi} k^2 \,\bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
  \, \cdot \; a_0\, (2\cdot 0 - 1) \, (k|x|)^{2 \cdot 0 - 2} .
  \end{alignedat}
\end{align}
For the last component, we have the following identity:
\begin{align*}
  A_1 = 
  & - \frac{1}{2\pi} k\, \bigg( 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \bigg)  \\
  &\,\cdot\, \bigg\{ 
  \; \mathlarger{\sum}_{l = 1}^\infty \; b_l \, (k|x|)^{2l - 1} \, C_l 
  + \mathlarger{\sum}_{l = 1}^\infty \; b_l \, (k|x|)^{2l - 1} \log(k|x|) 
  + \mathlarger{\sum}_{l = 0}^\infty \; a_l \, (k|x|)^{2l - 1} 
  \bigg\}.
\end{align*}
In this case, the prefactor behaves like $\sqrt{|\lambda|} |x|^{-2}$.
Therefore, problematic terms only arise in the last two sums
\begin{align}
  \label{eq:P3}
  \tag{P3}
  \begin{alignedat}{1}
%    &- \frac{1}{2\pi} k\, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)
%  \, \cdot \mathlarger{\sum}_{l = 1}^1 \; b_l \, (k|x|)^{2l - 1} \log(k|x|)  \\
%    &- \frac{1}{2\pi} k\, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)
%  \, \cdot \mathlarger{\sum}_{l = 0}^0 \; a_l \, (k|x|)^{2l - 1} .
    %
    &- \frac{1}{2\pi} k\, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)
    \, \cdot \; b_1 \, (k|x|)^{2 \cdot 1 - 1} \log(k|x|)  \quad\text{and}\\[0.5em]
    &- \frac{1}{2\pi} k\, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)
  \, \cdot \; a_0 \, (k|x|)^{2 \cdot 0 - 1} .
  \end{alignedat}
\end{align}
Note that we get from definition \eqref{defnConst}
\begin{align*}
  a_0 = 1, \qquad c_1 = b_1 = - \frac{1}{2}.
\end{align*}
Therefore, we can already see that the logarithmic terms in the sum $\eqref{eq:P2} + \eqref{eq:P3}$ cancel.
Next, we observe that if we take the sum over the problematic terms \eqref{eq:P1}, \eqref{eq:P2} and \eqref{eq:P3} and subtract $\partial_\beta \partial_\alpha \partial_\gamma G(x; 0)$, the result is $0$ which is easily seen by grouping the terms having the same power of $|x|$:
\begin{samepage}
\begin{align*}
  &\hspace{-2cm}\mathrm{(P1)} + \mathrm{(P2)} + \mathrm{(P3)} - \partial_\beta \partial_\alpha \partial_\gamma G(x; 0) \\[0.5em]
  = &- \frac{1}{\pi}\, \frac{x_\alpha x_\beta x_\gamma}{|x|^6} \\[0.5em]
  %
    %&+ \frac{1}{4\pi} k^2 \bigg(\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3 \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \bigg) \log(k|x|)  \\[0.5em]
    %
    &+ \frac{1}{2\pi} \, \bigg(\,\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^4} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^6} \, \bigg) \\[0.5em]
    %
    %&+ \frac{1}{4\pi} k^2 \bigg( 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^2} \bigg) \,  \log(k|x|)  \\[0.5em]
    &- \frac{1}{2\pi} \, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^6} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^4} \, \bigg)\\[0.5em]
    %
   &- \frac{1}{\pi}\, \frac{\delta_{\beta \gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^4} 
     + \frac{4}{\pi}\, \frac{x_\alpha x_\beta x_\gamma}{|x|^6} 
     \quad=\quad 0 \,. 
\end{align*}
This completes the proof of Theorem~\ref{lem:HelmholtzLaplaceDifference}. \hfill$\qed$
\end{samepage}


%\newpage
\section*{A.2\quad Proof of Theorem \ref{thm:differenceFundamentalSolutionStokes} for $d = 2$}
\markboth{APPENDIX}{A.2.\quad PROOF OF THEOREM \ref*{thm:differenceFundamentalSolutionStokes} FOR $d = 2$}
\addcontentsline{toc}{section}{A.2\quad Proof of Theorem \ref*{thm:differenceFundamentalSolutionStokes} for $d = 2$}
\label{sec:A2}

In this section, we show that, for all $\lambda \in \Sigma_\theta$, $\theta \in (0,\pi/2)$ and $x \in \R^2\setminus \{0\}$ satisfying $|\lambda| |x|^2 \leq (1/2)$, we have
\begin{align*}
  \Big|\, \nabla_x \big\{ \Gamma(x; \lambda) - \Gamma(x; 0) \big\} \, \Big| \leq C\, |\lambda| |x| \, \big|\log(|\lambda| |x|^2)\,\big|\,,
\end{align*}
where $C > 0$ depends only on $\theta$.
The means and the strategy to prove this estimate are similar to the procedure in the previous section.
In addition to the derivatives that were calculated at the beginning of this appendix, we will furthermore need the first partial derivatives for the matrix of fundamental solutions of the Stokes problem:
\begin{align*}
    \Gamma_{\alpha\beta}(x; 0) 
    &= \frac{1}{4\pi} \bigg\{ \,- \delta_{\alpha\beta} \log(|x|) + \frac{x_\alpha x_\beta}{|x|^2} \,\bigg\}
    \\[0.5em]
  \partial_\gamma \Gamma_{\alpha\beta} (x; 0) 
  &= \frac{1}{4 \pi}\, \bigg(\, \frac{\delta_{\alpha\gamma} x_\beta + \delta_{\beta\gamma} x_\alpha  - \delta_{\alpha\beta} x_\gamma}{|x|^2}  \, \bigg) 
  - \frac{1}{2\pi}\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} .
\end{align*}
Now consider the difference
\begin{align*}
  &\partial_\gamma \Gamma_{\alpha\beta}(x; \lambda) - \partial_\gamma \Gamma_{\alpha\beta}(x; 0) \\
  &\qquad=\partial_\gamma G(x; \lambda) \,\delta_{\alpha\beta}
  + \frac{1}{k^2} \, \partial_\beta \partial_\alpha \partial_\gamma \Big\{ G(x; \lambda) - G(x; 0) \Big\} 
  - \partial_\gamma \Gamma_{\alpha\beta}(x; 0) \eqqcolon B_1 + B_2 + B_3 \,,
\end{align*}
where we introduced the variables $B_i$, $i = 1,\dots,3$, for the sake of readability.
As in the previous section, we will study the terms $B_i$ independently and extract the terms that do not exhibit the desired behavior.
For $B_1$, we have
\begin{align*}
  B_1 = -\frac{1}{2\pi} k\;  \frac{\delta_{\alpha\beta} x_\gamma}{|x|}  \;
  \cdot \,\bigg\{ \;
  \mathlarger{\sum}_{l = 1}^\infty \; b_l \, (k|x|)^{2l - 1} \, C_l 
  &+ \mathlarger{\sum}_{l = 1}^\infty \; b_l \, (k|x|)^{2l - 1} \log(k|x|) \; \\
  &+  \mathlarger{\sum}_{l = 0}^\infty \; a_l \, (k|x|)^{2l - 1} \; \bigg\}.
\end{align*}
In this expression, we only detect one problematic term, namely the term corresponding to $l = 0$ in the last sum
\begin{align}
  \label{eq:Q1}
  \tag{Q1}
  %-\frac{1}{2\pi} k\;  \frac{\delta_{\alpha\beta} x_\gamma}{|x|}\, \mathlarger{\sum}_{l = 0}^0 \; a_l \, (k|x|)^{2l - 1} .
  -\frac{1}{2\pi} k\;  \frac{\delta_{\alpha\beta} x_\gamma}{|x|}\, \cdot \; a_0 \, (k|x|)^{2 \cdot 0 - 1} .
\end{align}
For the expression $B_2$, we have  
\begin{align*}
  B_2 = k^{-2} \, \Big( \, A_3 + A_2 + A_1 - \partial_\beta\partial_\alpha\partial_\gamma G(x; 0)\, \Big) \eqqcolon A_3' + A_2' + A_1'\,,
\end{align*}
with the variables $A_i$, $i = 1, \dots, 3$, which were introduced in the previous section. 
We will now list the problematic terms for $A_i'$ where we will already take into account the cancellations from the previous section.
This will result on the one hand in additional terms and on the other hand in subsequent terms in the same sums compared to \eqref{eq:P1}, \eqref{eq:P2} and \eqref{eq:P3}.

For $A_3'$, we see that the following term does not meet the desired behavior:
\begin{align}
  \label{eq:Q2}
  \tag{Q2}
  \begin{alignedat}{1}
  %&- \frac{1}{2 \pi } k \, \frac{x_\alpha x_\beta x_\gamma}{|x|^3} \; \mathlarger{\sum}_{l = 1}^1 \; c_l \, (k|x|)^{2l - 3}. 
  &- \frac{1}{2 \pi } k \, \frac{x_\alpha x_\beta x_\gamma}{|x|^3} \; \cdot \; c_1 \, (k|x|)^{2 \cdot 1 - 3}. 
  %&- \frac{1}{2 \pi } k \frac{x_\alpha x_\beta x_\gamma}{|x|^3}  \; \mathlarger{\sum}_{l = 1}^{1} \; a_l (2l - 1) (2l - 2) \, (k|x|)^{2l - 3} 
  \end{alignedat}
\end{align}
For $A_2'$, we see that, compared to $A_2$, every summand is problematic which after cancellation leads to:
\begin{align}
  \label{eq:Q3}
  \tag{Q3}
  \begin{alignedat}{1}
%  &- \frac{1}{2\pi} \, \bigg(\,\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
%  \,\cdot \mathlarger{\sum}_{l = 1}^1 \; c_l \, (k|x|)^{2l - 2} \, C_l \\
%  %
%  %&- \frac{1}{2\pi} \bigg(\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3 \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \bigg) 
%  %\cdot \mathlarger{\sum}_{l = 1}^1 \; c_l \, (k|x|)^{2l - 2} \log(k|x|) \\
%  %
%  &- \frac{1}{2\pi} \, \bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
%  \,\cdot \mathlarger{\sum}_{l = 1}^1 \; b_l \, (k|x|)^{2l - 2}  \\
%  %
%  &- \frac{1}{2\pi} \, \bigg(\,\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
%  \,\cdot \mathlarger{\sum}_{l = 1}^1 \; a_l (2l - 1) \, (k|x|)^{2l - 2} .
    %
  &- \frac{1}{2\pi} \, \bigg(\,\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
    \,\cdot \; c_1 \, (k|x|)^{2\cdot 1 - 2} \, C_1 \\[1.0em]
  %
  %&- \frac{1}{2\pi} \bigg(\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3 \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \bigg) 
  %\cdot \mathlarger{\sum}_{l = 1}^1 \; c_l \, (k|x|)^{2l - 2} \log(k|x|) \\
  %
  &- \frac{1}{2\pi} \, \bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
    \,\cdot \; b_1 \, (k|x|)^{2 \cdot 1 - 2}  \\[1.0em]
  %
  &- \frac{1}{2\pi} \, \bigg(\,\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
  \,\cdot \; a_1 (2 \cdot 1 - 1) \, (k|x|)^{2 \cdot 1 - 2} .
  \end{alignedat}
\end{align}
The same holds for the expression $A_1'$:
\begin{align}
  \label{eq:Q4}
  \tag{Q4}
  \begin{alignedat}{1}
%    &- \frac{1}{2\pi} \, \frac{1}{k} \, \bigg(\, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)  
%   \,\cdot \mathlarger{\sum}_{l = 1}^1 \; b_l \, (k|x|)^{2l - 1} \, C_l  \\
%    %
%    %&- \frac{1}{2\pi} \, \frac{1}{k} \Big( 3 \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \Big)  
%  %\mathlarger{\sum}_{l = 1}^1 \; b_l \, (k|x|)^{2l - 1} \log(k|x|) \\
%    %
%    &- \frac{1}{2\pi} \, \frac{1}{k} \, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)  
%  \,\cdot \mathlarger{\sum}_{l = 1}^1 \; a_l \, (k|x|)^{2l - 1} .
%    %
    &- \frac{1}{2\pi} \, \frac{1}{k} \, \bigg(\, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)  
    \,\cdot \; b_1 \, (k|x|)^{2 \cdot 1 - 1} \, C_1  \\[1.0em]
    %
    %&- \frac{1}{2\pi} \, \frac{1}{k} \Big( 3 \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \Big)  
  %\mathlarger{\sum}_{l = 1}^1 \; b_l \, (k|x|)^{2l - 1} \log(k|x|) \\
    %
    &- \frac{1}{2\pi} \, \frac{1}{k} \, \bigg( \, 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^5} - \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha\gamma} x_\beta + \delta_{\alpha\beta} x_\gamma}{|x|^3} \, \bigg)  
  \,\cdot \; a_1 \, (k|x|)^{2 \cdot 1 - 1} .
  \end{alignedat}
\end{align}
Now it is time to sum all problematic terms, expand the variables and add the term $B_3$. 
As before, all the terms will cancel which will then prove our initial claim. 
With
\begin{align*}
  a_0 = 1, \quad a_1 = -\frac{1}{4} \quad\text{and}\quad c_1 = b_1 =  -\frac{1}{2},
\end{align*}
we see that already within the sum \eqref{eq:Q3} + \eqref{eq:Q4} a lot of terms cancel which leaves us with:
\begin{align*}
  \label{eq:Q33}
  \tag{Q3'}
  &- \frac{1}{2\pi} \, \bigg(\, \frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
  \, \cdot b_1 \, (k|x|)^{2\cdot 1 - 2}.
\end{align*}
Now let us consider the expression $\eqref{eq:Q1} + \eqref{eq:Q2} + \eqref{eq:Q33} + B_3$:
\begin{align*}
  &-\frac{1}{2\pi} \,  \frac{\delta_{\alpha\beta} x_\gamma}{|x|^2} 
  + \frac{1}{4 \pi }\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} 
  %
  + \frac{1}{4\pi} \, \bigg(\,\frac{\delta_{\beta\gamma} x_\alpha + \delta_{\alpha \gamma} x_\beta + \delta_{\alpha \beta} x_\gamma}{|x|^2} - 3\, \frac{x_\alpha x_\beta x_\gamma}{|x|^4} \, \bigg) 
   \\[0.5em]
 &\quad - \frac{1}{4 \pi} \, \bigg(\, \frac{\delta_{\alpha\gamma} x_\beta + \delta_{\gamma\beta} x_\alpha  - \delta_{\alpha\beta} x_\gamma}{|x|^2} \, \bigg) 
  + \frac{1}{2\pi}\,  \frac{x_\alpha x_\beta x_\gamma}{|x|^4}
  \quad = \quad 0\, .
\end{align*}
As all problematic terms add up to zero, we have proven the initial claim.\hfill$\qed$

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