diff --git a/theory/Multipoles/potential_derivatives.tex b/theory/Multipoles/potential_derivatives.tex
index c62f83df642f1eb7542cdef04aaab1eeb1635d50..9a1d23384bdc5b67f62b845a45672a47b4030893 100644
--- a/theory/Multipoles/potential_derivatives.tex
+++ b/theory/Multipoles/potential_derivatives.tex
@@ -74,6 +74,216 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H)
% \chi^{(5)}(r, r_s) &= \frac{32}{r_s^5} \left(240\alpha(x)^6 - 720\alpha(x)^5 + 780\alpha(x)^4 - 360\alpha(x)^3 + 62\alpha(x)^2 - 2\alpha(x) \right) \nonumber
% \end{align}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{align}
+ \mathsf{\tilde{D}}_{1}(r, r_s, H) = % D_tilde_tilde_1 = D_tilde_1
+ \left\lbrace\begin{array}{rcl}
+ f(u)\times H^{-1} & \mbox{if} & u < 1,\\
+ %r^{-1} & \mbox{if} & u \geq 1,
+ \chi \times r^{-1} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
+ r^{-1} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
+ \end{array}
+ \right.\nonumber
+\end{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{align}
+ \mathsf{\tilde{D}}_{2}(r, r_s, H) = % D_tilde_tilde_3 = D_tilde_3 * r
+ \left\lbrace\begin{array}{rcl}
+ f'(u)\times H^{-2}& \mbox{if} & u < 1,\\
+ %-1 \times r^{-3} & \mbox{if} & u \geq 1,
+ \left(r\chi' - \chi\right) \times r^{-2} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
+ -1 \times r^{-2} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
+ \end{array}
+ \right.\nonumber
+\end{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{align}
+ \mathsf{\tilde{D}}_{3}(r, r_s, H) = % D_tilde_tilde_5 = D_tilde_5 * r^2
+ \left\lbrace\begin{array}{rcl}
+ \left(f''(u) - u^{-1}f'(u)\right)\times H^{-3}& \mbox{if} & u < 1,\\
+ %3\times r^{-5} & \mbox{if} & u \geq 1,
+ \left(r^2\chi'' - 3r\chi' + 3\chi \right)\times r^{-3} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
+ 3 \times r^{-3} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
+ \end{array}
+ \right.\nonumber
+\end{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{align}
+ \mathsf{\tilde{D}}_{4}(r, r_s, H) = % D_tilde_tilde_7 = D_tilde_7 * r^3
+ \left\lbrace\begin{array}{rcl}
+ \left(f^{(3)}(u)-3u^{-1}f''(u)+3u^{-2}f'(u)\right)\times H^{-4} & \mbox{if} & u < 1,\\
+ %-15\times r^{-7} & \mbox{if} & u \geq 1,
+ \left(r^3\chi^{(3)} - 6r^2\chi''+15r\chi'-15\chi\right) \times r^{-4} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
+ -15 \times r^{-4} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
+ \end{array}
+ \right.\nonumber
+\end{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{align}
+ \mathsf{\tilde{D}}_{5}(r, r_s, H) = % D_tilde_tilde_9 = D_tilde_9 * r^4
+ \left\lbrace\begin{array}{rcl}
+ \left(f^{(4)}(u)-6u^{-1}f^{(3)}(u)+15u^{-2}f''(u)-15u^{-3}f'(u)\right)\times H^{-5}& \mbox{if} & u < 1,\\
+ %105\times r^{-9} & \mbox{if} & u \geq 1.
+ \left(r^4\chi^{(4)} - 10r^3\chi^{(3)} + 45r^2\chi'' - 105r\chi' + 105\chi \right) \times r^{-5} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
+ 105 \times r^{-5} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
+ \end{array}
+ \right.\nonumber
+\end{align}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{align}
+ \mathsf{\tilde{D}}_{6}(r, r_s, H) = % D_tilde_tilde_11 = D_tilde_11 * r^4
+ \left\lbrace\begin{array}{rcl}
+ \left(f^{(5)}(u) -10u^{-1}f^{(4)}(u) +45u^{-2}f^{(3)} -105u^{-3}f''(u) + 105u^{-4}f'(u)\right)\times H^{-6}& \mbox{if} & u < 1,\\
+ %-945\times r^{-11} & \mbox{if} & u \geq 1.
+ \left(r^5\chi^{(5)} - 15r^4\chi^{(4)} + 105r^3\chi^{(3)} - 420r^2\chi'' + 945r \chi' - 945\chi\right) \times r^{-6} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
+ -945\times r^{-6} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
+ \end{array}
+ \right.\nonumber
+\end{align}
+In the case $u<1$ and using $f(u)$ given by \ref{eq:fmm:potential}, we can simplify the expressions to get:
+\begin{align}
+ \mathsf{\tilde{D}}_{1} &= (-3u^7 + 15u^6 - 28u^5 + 21u^4 - 7u^2 + 3) \times H^{-1}, \nonumber \\
+ \mathsf{\tilde{D}}_{2} &= (-21u^6 + 90u^5 - 140u^4 + 84u^3 - 14u) \times H^{-2}, \nonumber \\
+ \mathsf{\tilde{D}}_{3} &= (-105u^5 + 360u^4 - 420u^3 + 168u^2) \times H^{-3}, \nonumber \\
+ \mathsf{\tilde{D}}_{4} &= (-315u^4 + 720u^3 - 420u^2) \times H^{-4}, \nonumber \\
+ \mathsf{\tilde{D}}_{5} &= (-315u^3 + 420u) \times H^{-5}, \nonumber \\
+ \mathsf{\tilde{D}}_{6} &= (315u^2 - 1260) \times H^{-6}. \nonumber
+\end{align}
+We can now write out all the derivatives used in the M2L and M2P kernels:
+\begin{align}
+ \mathsf{D}_{000}(\mathbf{r}) = \varphi (\mathbf{r}, r_s, H) =
+ \mathsf{\tilde{D}}_{1} \nonumber
+\end{align}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\noindent\rule{6cm}{0.4pt}
+\begin{align}
+ \mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right) \mathsf{\tilde{D}}_{2} \nonumber
+\end{align}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\noindent\rule{6cm}{0.4pt}
+\begin{align}
+\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \varphi (\mathbf{r}, r_s, H) =
+\left(\frac{r_x}{r}\right)^2 \mathsf{\tilde{D}}_{3} + \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{2}\nonumber
+\end{align}
+
+\begin{align}
+\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r}, r_s, H) =
+\left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{3} \nonumber
+\end{align}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\noindent\rule{6cm}{0.4pt}
+\begin{align}
+\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^3 \mathsf{\tilde{D}}_{4}
+ + 3 \left(\frac{r_x}{r}\right) \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{3} \nonumber
+\end{align}
+
+\begin{align}
+\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^2 r_y} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{4} + \left(\frac{r_y}{r}\right) \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{3} \nonumber
+\end{align}
+
+\begin{align}
+\mathsf{D}_{111}(\mathbf{r}) = \frac{\partial^3}{\partial r_x\partial r_y\partial r_z} \varphi (\mathbf{r}, r_s, H) =
+\left(\frac{r_x}{r}\right)\left(\frac{r_y}{r}\right)\left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{4} \nonumber
+\end{align}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\noindent\rule{6cm}{0.4pt}
+\begin{align}
+ \mathsf{D}_{400}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^4}
+ \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^4 \mathsf{\tilde{D}}_{5}+
+ 6\left(\frac{r_x}{r}\right)^2 \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4} +
+ 3 \left(\frac{1}{r}\right)^2 \mathsf{\tilde{D}}_{3}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{310}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^3
+ \partial r_y} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^3 \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{4} +
+ 3 \left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right) \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{3}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{220}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^2
+ \partial r_y^2} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right)^2 \mathsf{\tilde{D}}_{5} +
+ \left(\frac{r_x}{r}\right)^2 \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4} +
+ \left(\frac{r_y}{r}\right)^2 \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4} +
+ \left(\frac{1}{r}\right)^2 \mathsf{\tilde{D}}_{3}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{211}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^2
+ \partial r_y \partial r_z} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^2\left(\frac{r_y}{r}\right)\left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{5} +
+ \left(\frac{r_y}{r}\right)\left(\frac{r_z}{r}\right)\left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4}
+ \nonumber
+\end{align}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\noindent\rule{6cm}{0.4pt}
+\begin{align}
+ \mathsf{D}_{500}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^5}
+ \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^5 \mathsf{\tilde{D}}_{6} +
+ 10\left(\frac{r_x}{r}\right)^3\left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
+ 15\left(\frac{r_x}{r}\right)\left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{410}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^4
+ \partial r_y} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^4 \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{6} +
+ 6 \left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right)\left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{5} +
+ 3 \left(\frac{r_y}{r}\right) \left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{320}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
+ \partial r_y^2} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^3 \left(\frac{r_y}{r}\right)^2 \mathsf{\tilde{D}}_{6} +
+ \left(\frac{r_x}{r}\right)^3 \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
+ 3 \left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right)^2 \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
+ 3 \left(\frac{r_x}{r}\right) \left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{311}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
+ \partial r_y \partial r_z} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^3 \left(\frac{r_y}{r}\right) \left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{6} +
+ 3 \left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right) \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5}
+ \nonumber
+\end{align}
+
+\begin{align}
+ \mathsf{D}_{221}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^2
+ \partial r_y^2 \partial r_z} \varphi (\mathbf{r}, r_s, H) =
+ \left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right)^2 \left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{6} +
+ \left(\frac{r_x}{r}\right)^2 \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
+ \left(\frac{r_y}{r}\right)^2 \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
+ \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
+ \nonumber
+\end{align}
+
+\begin{comment}
+\noindent\rule{12cm}{1pt}\\
+Old version \\
+\noindent\rule{12cm}{1pt}
+
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{1}(r, r_s, H) =
@@ -266,6 +476,7 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
\nonumber
\end{align}
+\end{comment}
@@ -285,94 +496,3 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{comment}
-
-\noindent\rule{6cm}{0.4pt}
-
-\begin{align}
-\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r},H) =
-\left\lbrace\begin{array}{rcl}
--\frac{r_x}{H^3} \left(21u^5 - 90u^4 + 140u^3 - 84u^2 + 14\right) & \mbox{if} & u < 1,\\
--\frac{r_x}{r^3} & \mbox{if} & u \geq 1,
-\end{array}
-\right.\nonumber
-\end{align}
-
-\noindent\rule{6cm}{0.4pt}
-
-\begin{align}
-\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \varphi (\mathbf{r},H) =
-\left\lbrace\begin{array}{rcl}
-\frac{r_x^2}{H^5}\left(-105u^3+360u^2-420u+168\right) -
-\frac{1}{H^3} \left(21u^5 - 90u^4 + 140u^3 - 84u^2 + 14\right) & \mbox{if} & u < 1,\\
-3\frac{r_x^2}{r^5} - \frac{1}{r^3} & \mbox{if} & u \geq 1,
-\end{array}
-\right.\nonumber
-\end{align}
-
-\begin{align}
-\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r},H) =
-\left\lbrace\begin{array}{rcl}
-\frac{r_xr_y}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
-3\frac{r_xr_y}{r^5} & \mbox{if} & u \geq 1,
-\end{array}
-\right.\nonumber
-\end{align}
-
-\noindent\rule{6cm}{0.4pt}
-
-\begin{align}
-\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) =
-\left\lbrace\begin{array}{rcl}
--\frac{r_x^3}{H^7} \left(315u - 720 + 420u^{-1}\right) +
-\frac{3r_x}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
--15\frac{r_x^3}{r^7} + 9 \frac{r_x}{r^5} & \mbox{if} & u \geq 1,
-\end{array}
-\right.\nonumber
-\end{align}
-
-\begin{align}
-\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) =
-\left\lbrace\begin{array}{rcl}
--\frac{r_x^2r_y}{H^7} \left(315u - 720 + 420u^{-1}\right) +
-\frac{r_y}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
--15\frac{r_x^2r_y}{r^7} + 3 \frac{r_y}{r^5} & \mbox{if} & u \geq 1,
-\end{array}
-\right.\nonumber
-\end{align}
-
-
-\begin{align}
-\mathsf{D}_{111}(\mathbf{r}) = \frac{\partial^3}{\partial r_x\partial r_y\partial r_z} \varphi (\mathbf{r},H) =
-\left\lbrace\begin{array}{rcl}
--\frac{r_xr_yr_z}{H^7} \left(315u - 720 + 420u^{-1}\right) & \mbox{if} & u < 1,\\
--15\frac{r_xr_yr_z}{r^7} & \mbox{if} & u \geq 1,
-\end{array}
-\right.\nonumber
-\end{align}
-
-\noindent\rule{6cm}{0.4pt}
-
-\begin{align}
- \mathsf{D}_{400}(\mathbf{r}) &=
- \nonumber
-\end{align}
-
-\begin{align}
- \mathsf{D}_{310}(\mathbf{r}) &=
- \nonumber
-\end{align}
-
-\begin{align}
- \mathsf{D}_{220}(\mathbf{r}) &=
- \nonumber
-\end{align}
-
-\begin{align}
- \mathsf{D}_{211}(\mathbf{r}) &=
- \nonumber
-\end{align}
-
-\end{comment}