updating math documentation, and fixing some typos in laplace fmm doc…

…umentation

docs/math.rst

@@ -5,6 +5,92 @@ of source locations and let $t_{i} \in \mathbb{R}^{2}$ denote a collection
 of target locations. 
 
 
+Laplace FMM
+*************
+The Laplace FMM comes in three varieties
+
+* rfmm2d: Charges, dipole strengths, potentials, their
+  gradients, and hessians are all double precision 
+* lfmm2d: All of the above quantities are double complex
+* cfmm2d: Same as lfmm2d except that there are no dipole orientation vectors,
+  and the dipole kernel is given by $1/(zt_{i} - zx_{j})$ where $zt_{i}, zx_{j}$
+  are the complex versions of the source and target locations given by
+  $zt_{i} = t_{i}(1) + i*t_{i}(2)$, and $zx_{j} = x_{j}(1) + i*y_{j}(2)$. 
+  Moreover, the gradients, and hessians
+  in this case are derivatives with respect to $z$, and in a slight abuse of
+  notation, we set $d/dz log(|z|) = 1/z$
+
+
+Laplace FMM (rfmm2d)
+---------------------
+Let $c_{j} \in \mathbb{R}$, 
+$j=1,2,\ldots N$, 
+denote a collection of charge strengths, $v_{j} \in \mathbb{R}$,
+$j=1,2,\ldots N$, 
+denote a collection of dipole strengths, and $d_{j} \in \mathbb{R}^{2}$,
+$j=1,2,\ldots N$, denote the corresponding dipole orientation vectors.
+
+The Laplace FMM (rfmm2d) computes 
+the potential $u(x) \in \mathbb{R}$ and the its gradient $\nabla u(x) \in
+\mathbb{R}^{2}$
+given by
+
+.. math::
+   :label: rlap_nbody
+
+    u(x) = \sum_{j=1}^{N} c_{j} log(\|x-x_{j}\|) - v_{j} d_{j}\cdot \nabla log(\|x-x_{j}\|)  \, , 
+
+at the source and target locations. When $x=x_{j}$, the term
+corresponding to $x_{j}$ is dropped from the sum.
+
+
+Laplace FMM (lfmm2d)
+---------------------
+Let $c_{j} \in \mathbb{C}$, 
+$j=1,2,\ldots N$, 
+denote a collection of charge strengths, $v_{j} \in \mathbb{C}$,
+$j=1,2,\ldots N$, 
+denote a collection of dipole strengths, and $d_{j} \in \mathbb{R}^{2}$,
+$j=1,2,\ldots N$, denote the corresponding dipole orientation vectors.
+
+The Laplace FMM (rfmm2d) computes 
+the potential $u(x) \in \mathbb{C}$ and the its gradient $\nabla u(x) \in
+\mathbb{C}^{2}$
+given by
+
+.. math::
+   :label: llap_nbody
+
+    u(x) = \sum_{j=1}^{N} c_{j} log(\|x-x_{j}\|) - v_{j} d_{j}\cdot \nabla log(\|x-x_{j}\|)  \, , 
+
+at the source and target locations. When $x=x_{j}$, the term
+corresponding to $x_{j}$ is dropped from the sum.
+
+
+Laplace FMM (cfmm2d)
+---------------------
+Let $c_{j} \in \mathbb{C}$, 
+$j=1,2,\ldots N$, 
+denote a collection of charge strengths, and $v_{j} \in \mathbb{C}$,
+$j=1,2,\ldots N$, 
+denote a collection of dipole strengths.
+
+The Laplace FMM (cfmm2d) computes 
+the potential $u(z) \in \mathbb{C}$ and the its gradient $d/dz u(z) \in
+\mathbb{C}$
+given by
+
+.. math::
+   :label: clap_nbody
+
+    u(z) = \sum_{j=1}^{N} c_{j} log(\|z-zx_{j}\|) - \frac{v_{j}}{z-zx_{j}}  \, , 
+
+at the source and target locations. When $z=zx_{j}$, the term
+corresponding to $zx_{j}$ is dropped from the sum. Here
+$z = x(1) + i x(2)$, and $zx_{j} = x_{j}(1) + ix_{j}(2)$, are the complex
+numbers corresponding to $x$ and $x_{j}$ respectively.
+
+
 Helmholtz FMM
 *************
 Let $c_{j} \in \mathbb{C}$, 
@@ -29,6 +115,16 @@ at the source and target locations, where $H_{0}^{(1)}$ is the Hankel function
 of the first kind of order $0$. When $x=x_{j}$, the term
 corresponding to $x_{j}$ is dropped from the sum.
 
+
+Biharmonic FMM
+***************
+
+Modified Biharmonic FMM
+************************
+
+Stokes FMM
+************
+
 Vectorized versions   
 *******************
 The vectorized versions of the Helmholtz FMM, 

src/laplace/cfmm2d.f

@@ -1,4 +1,4 @@
-cc Copyright (C) 2018-2019: Le=",nlevelsslie Greengard, Zydrunas Gimbutas, 
+cc Copyright (C) 2018-2019: Leslie Greengard, Zydrunas Gimbutas, 
 cc and Manas Rachh
 c     c Contact: [email protected]
 cc

src/laplace/cfmm2dwrap.f

@@ -23,7 +23,7 @@
 c
 c   l2d: charge and dipstr are complex valued, x in \R^2
 c
-c   \phi(x_i) = \sum_{j\ne i} charge_j log|x_i-x_j}
+c   \phi(x_i) = \sum_{j\ne i} charge_j log|x_i-x_j|
 c   + dipstr_j/x_i - x_j
 c
 c

src/laplace/rfmm2dwrap.f

@@ -28,7 +28,7 @@
 c
 c   l2d: charge and dipstr are real valued, x in \R^2
 c
-c   \phi(x_i) = \sum_{j\ne i} charge_j log|x_i-x_j}
+c   \phi(x_i) = \sum_{j\ne i} charge_j log|x_i-x_j|
 c   + dipstr_j/x_i - x_j
 c
 c