Integrate a gaussian distrbution over a sphere surface I want to integrate an off-center guassian distribtuion function over the surface of a sphere with radius of R. Here is the function:
$$ f(x,y,z) = A \exp(-\frac{(x-x_{s})^2+(y-y_{s})^2}{d^2})$$
$x_s, y_s, R, d$ and $A$ are all constant paramaters, and I can't replace them with numbers. The goal is to have a function based on all these parameters, that I can later use in my model.
In Mathemtica. I can solve a centric gussian but as it gets off-center(as I put $x_s$ and $y_s$) it can not be sovled.
To be able to easily integrate over surface of sphere I transform the equations to spherical coordinates and I have:
$$ \int_{0}^{2\pi} \int_{0}^{\pi} A R^2 \sin(\phi)\exp( -2 \frac{ (R \sin(\phi) \cos(\theta)-X_s)^2 + (R \sin(\phi) \sin(\theta)-Y_s)^2 }{d^2}) d\phi d\theta$$
But still I can not sovle this with MATLAB or Mathematica.
And I believe I can't solve it numerically because since I have parameters in there, solving it numerically would mean that for each element of integration I will have a different exp term. So I will just have a function with 1000 exponential terms (assuming I divide surface to 1000 elements) and that would be not practically useful.
Am I doing sth wrong? Any suggestions please?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\left\{\begin{array}{l}
\ds{\on{f}\pars{x,y,z} \equiv
A \exp\pars{-\,{\bracks{x - x_{s}}^{2} +
\bracks{y - y_{s}}^{2} \over d^{2}}}}
\\[1mm]
\ds{\bbox[5px,#ffd]{\iint_{\mathcal{S}}\on{f}\pars{x,y,z}
\,\dd S}:\ {\LARGE ?}}.
\\[1mm]
\mathcal{S} \equiv \braces{\pars{x,y,z}\ \mid\ 
x^{2} + y^{2} + z^{2} = R^{2}\,,\ R > 0}
\end{array}\right.$

Lets
$\ds{\vec{r} \equiv x\,\hat{x} + y\,\hat{y} + z\,\hat{z}\quad}$ and
$\ds{\quad\vec{r}_{s} \equiv x_{s}\,\hat{x} + y_{s}\,\hat{y} + 0\,\hat{z}}$. Then,
\begin{align}
&\bbox[5px,#ffd]{\iint_{\mathcal{S}}\on{f}\pars{x,y,z}
\,\dd S} =
\iint_{S}A\exp\pars{-\,{\verts{\vec{r} - \vec{r}_{s}}^{2} - z^{2} \over d^{2}}}\,\dd S
\\[5mm] = &\
A\int_{0}^{2\pi}
\\ &\
\!\!\!\!\!\int_{0}^{\pi}\!\!\!\!\!
\exp\pars{-\,{\bracks{R^{2} -2Rr_{s}\cos\pars{\theta} + r_{s}^{2}} - R^{2}\cos^{2}\pars{\theta} \over d^{2}}}\ \times
\\ &\ \phantom{\int_{0}^{2\pi}\int_{0}^{\pi}}
R^{2}\sin\pars{\theta}\,\dd\theta\,\dd\phi
\\[5mm] = &\
2\pi AR^{2}\exp\pars{-\,{R^{2} + r_{s}^{2} \over d^{2}}}\ \times
\\ &\
\int_{-1}^{1}
\exp\pars{-\,{-2Rr_{s}\,\xi - R^{2}\xi^{2} \over d^{2}}}
\,\dd\xi
\\[5mm] = &\
\left.2\pi ARd
\exp\pars{-\,{R^{2} + r_{s}^{2} \over d^{2}}}
\int_{-d/R}^{d/R}
\expo{\xi^{2}\ +\ 2\overline{r}_{s}\,\xi}
\,\dd\xi\,\right\vert_{\ \overline{r}_{s}\ =\ r_{s}\,/d}
\\[5mm] = &\
2\pi ARd
\exp\pars{-\,{R^{2} + r_{s}^{2} \over d^{2}}}\ \times
\\ &\
\braces{%
{1 \over 2}\expo{-\overline{r}_{s}^{2}}\root{\pi}
\bracks{\on{erfi}\pars{{d \over R} - \overline{r}_{s}} +
\on{erfi}\pars{{d \over R} + \overline{r}_{s}}}}
\\[5mm] = &\
\pi^{3/2}ARd
\exp\pars{-\,{R^{2} + 2r_{s}^{2} \over d^{2}}}\ \times
\\ &\
\bracks{\on{erfi}\pars{{d \over R} - {r_{s} \over d}} +
\on{erfi}\pars{{d \over R} + {r_{s} \over d}}}
\end{align}
whith $\ds{r_{s} = \root{x_{s}^{2} + y_{s}^{2}}\quad}$ and
$\ds{\quad\on{erfi}\pars{z} \equiv
-\ic\on{erf}\pars{\ic z}}$.

$\ds{\overline{\underline{\mbox{Note that}}}}$
$\ds{\ \color{red}{\on{erfi}\pars{z} \in \mathbb{R}}}$ when
$\ds{\color{red}{z \in \mathbb{R}}}$.

$\ds{\on{\large erfi}\quad\mbox{and}\quad\on{\large erf}\quad}$ belong to the
Error Function Family.
