text:
\section{Introduction}\label{intro}\footnote{This paper is a revisited, completed and finalized version of two previous  articles: arXiv:0912.4684 and arXiv:1001.4114}
In the cases of electrostatic and magnetostatic fields of point-like dipoles, one has the well-known procedure of introducing  Dirac $\delta$-function terms for obtaining correct expressions of the electric and magnetic fields defined on the entire space.  The corresponding field expressions take the following form \cite{Jackson}:
\beqa\label{1.1}
\fl\;\;\;\;\;\;\Evec_{\pvec}(\rvec)=-\frac{1}{3\eps_0}\,\pvec\,\delta(\rvec)+\frac{1}{4\pi\eps_0}\left(\frac{3(\nuvec\cdot\pvec)\nuvec-\pvec}{r^3}\right)_{r\ne 0}=-\frac{1}{3\eps_0}\,\pvec\,\delta(\rvec) + \big(\Evec\big)_{r\ne 0}\ ,
\eeqa
where $\nuvec=\rvec/r$, and
\beqa\label{1.2}
\fl\;\;\;\;\;\;\Bvec_{\mvec}(\rvec)= \frac{2\mu_0}{3}\,\mvec\,\delta(\rvec)+\frac{\mu_0}{4\pi}\left(\frac{3(\nuvec\cdot\mvec)\nuvec-\mvec}{r^3}\right)_{r\ne 0}=
\frac{2\mu_0}{3}\,\mvec\,\delta(\rvec)+ \big(\Bvec\big)_{r\ne 0}\ .
\eeqa
In these equations, by $(\dots)_{r\ne 0}$ we understand an expression in which the derivatives are calculated supposing $r\ne 0$, representing some well-known expressions of the fields.
The expressions from equations \eref{1.1} and \eref{1.2} are introduced in Ref.\  \cite{Jackson} as  conditions of compatibility with the average value of the  electric or magnetic field  over a spherical domain containing all the charges or currents inside. Another procedure for introducing equations \eref{1.1} and \eref{1.2} is based on an extension of the derivative $\d_i\d_j/(1/r)$ to the entire space \cite{Frahm}:
\beqa\label{1.3}
\d_i\d_j\frac{1}{r}=\left(\frac{3\,\nu_i\nu_j\,-\,\delta_{ij}}{r^3}\right)_{r\ne 0}\,-\frac{4\pi}{3}\,\delta_{ij}\,\delta(\rvec)
\eeqa
and this procedure will be applied in the present paper.
A more pedagogical and suitable approach for understanding the origin of the difference between  the electric and magnetic cases  is done in Ref.\  \cite{Leung06}.  Refs.\  \cite{Werner} and \cite{Leung07}  contain  generalizations of the equations \eref{1.1} and \eref{1.2} to the dynamic case for  oscillating electric and magnetic dipoles.\\
 Section \ref{prelim} is dedicated to the reader less used with the tensorial formalism in handling multipolar expansions. For the informed reader, the section can be seen as an introduction to the notation and formulas used along the article.    In Sections \ref{estatic} and \ref{magnetost}  the results for the electric and magnetic fields  are presented in the static case. The dynamic case is treated  in Section \ref{dynamic}, and the  conclusions are outlined in  Section \ref{conclusion}. \\
 The formalism employed in this paper is a purely algebraic one. With a good understanding of the  definitions and notation presented in Section \ref{prelim}, we think the reader will be  able to verify every detail of the calculation  and, possibly, to search and find some more adequate versions.

\section{Preliminaries}\label{prelim}

\subsection{Definitions, notation and formulas from the tensorial formalism}
Let us consider a $n$-th order Cartesian tensor denoted by $\Tsft^{(n)}$ and characterized by the components
$\Tsf_{i_1\dots i_n},\;\;i_q=1,2,3$. These components are, in fact, the components of a vector in the $n$ tensorial
product of the Euclidean space $\mathbb{R}^3 $:
\beqan
\Tsft^{(n)}=\Tsf_{i_1\dots i_n}\,\evec_{i_1}\otimes\,\dots\,\otimes\evec_{i_n}=\Tsf_{i_1\dots i_n}\,
{\esft}_{i_1\dots i_n}\ ,
\eeqan
where $\evec_i$ are the unit vectors of a Cartesian basis in $\mathbb{R}^3$ and ${\esft}_{i_1\dots i_n} $ are the
unit vectors of the tensorial product space. Some vectors from the tensorial product of spaces
can be represented by the tensorial products of $n$ vectors from the different space factors $\mathbb{R}^3$.  Particularly, in the case of  identical factors, we employ the usual notation
\beqan
\bbox{a}^n=a_{i_1}\dots a_{i_n}{\esft}_{i_1\dots i_n}\ .
\eeqan
The vector $\bbox{a}$ can be the differential operator $\nablav=\evec_i\,\d_i$ and, in this case,
\beqan
\left(\nablav^n\right)={\esft}_{i_1\dots i_n}\,\d_{i_1}\dots \d_{i_n}\ .
\eeqan
Because of this simplified notation, for avoiding confusions, the Laplace operator  in the space $\mathbb{R}^3 $ is symbolized  by $\Delta$, and not by the frequent  $\nablav^2$. \\
We employ the following notation for the tensorial contractions:
\beqan
 ({\Atens}^{(n)}||{\Btens}^{(m)})_{i_1 \cdots i_{|n-m|}}
=\left\{\begin{array}{ll}
A_{i_1 \cdots i_{n-m}j_1 \cdots j_m}B_{j_1 \cdots j_m} & ,\; n>m\\
A_{j_1 \cdots j_n}B_{j_1 \cdots j_n} & ,\; n=m\\
A_{j_1 \cdots j_n}B_{j_1 \cdots j_n i_1 \cdots i_{m-n}} & ,\; n<m
\end{array} \right.\ .
\eeqan
A fully symmetric tensor $\ssft^{(n)}$, which here is called simply "symmetric", has a projection on the subspace
of symmetric and trace free ({\bf STF}) tensors. Up to a numerical factor, this projection will be represented by the tensor  $\Tcal(\ssft^{(n)})\equiv \bbox{\mathcal{S}}^{(n)}$. For  $n=2$, for example, we can write
\beqa\label{a.2}
\ssf_{ij}=\mathcal{S}_{ij}+\delta_{ij}\,\latens(\ssft^{(2)})\ .
\eeqa
The condition $\mathcal{S}_{ii}=0$ gives
 \beqa\label{a.3}
\lasf=\frac{1}{3}\ssf_{qq}\ .
\eeqa
In the case $n=3$:
\beqan
\ssf_{ijk}=\mathcal{S}_{ijk}+\delta_{\{ij}\lasf_{k\}}(\ssft^{(3)})\ ,
\eeqan
where  by the symbol $A_{\{i_1\dots i_n\}}$ is denoted the sum over all distinct permutations of the indexes.
The condition $\mathcal{S}_{iik}=0$ for $k=1,2,3$ implies
 \beqan
\lasf_i(\ssft^{(3)})=\frac{1}{5}\ssf_{qqi}\ .
 \eeqan

Though, maybe, only for a theoretical interest, let us consider  the general case for the {\bf STF} projection   of the symmetric tensor $\ssft^{(n)}$ defined up to a numerical factor by the equation
\beqan
\Tcal_{i_1\dots i_n}(\ssft^{(n)})\equiv \mathcal{S}_{i_1\dots i_n}=\ssf_{i_1\dots i_n}-\delta_{\{i_1i_2}\lasf_{i_3\dots i_n\}}(\ssft^{(n)})\ .
\eeqan
 $\latens^{(n-2)}\equiv\latens(\ssft^{(n)})$ is a symmetric tensor and is defined by the condition that $\Tcal^{(n)}$ is a trace-free
tensor. For low values of $n$, the ones of really practical interest, the components $\lasf_{i_1\dots i_{n-2}}$ can be calculated directly from the equation system representing the vanishing relations of all the partial traces of the tensor $\Tcal^{(n)}$. However, we mention here a general formula known from literature \cite{Thorne,App} which, with
the notation from the present paper, is written as
\beqan
&~&\fl\left[\Tcal\big[\ssft^{(n)}\big]\right]_{i_1\dots i_n}=
\suml^{[n/2]}_{m=0}\frac{(-1)^m(2n-1-2m)!!}{(2n-1)!!}\delta_{\{i_1i_2}\dots
\delta_{i_{2m-1}i_{2m}}\ssf^{(n:m)}_{i_{2m+1}\dots i_n\}} \ .
\eeqan
 The symbol $[\al]$ represents the integer part of $\al$ and $\ssf^{(n:m)}_{i_{2m+1}\dots i_n}$ denotes the components of the $(n-2\,m)$-th order tensor
obtained from $\ssft^{(n)}$ by contracting $m$ pairs of symbols $i$. This equation is known as the {\it detracer theorem} \cite{App}. As a consequence of this theorem, the components of the tensor $\latens^{(n-2)}$ are written as
\beqan
\fl\lasf_{i_1\dots i_{n-2}}\big[\ssft^{(n)}\big]=
\suml^{[n/2-1]}_{m=0}\frac{(-1)^m[2n-1-2(m+1)]!!}{(m+1)(2n-1)!!}
\delta_{\{i_1i_2}\dots \delta_{i_{2m-1}i_{2m}}\ssf^{(n:\,m+1)}_{i_{2m+1}\dots i_{n-2}\}}.
\eeqan
These formulas are useful for defining and processing the multipole expansions of the electrodynamic potentials and fields.
\subsection{Multipole expansion of the electromagnetic field in Cartesian coordinates}
The multipole expansions of the potentials in the Lorenz gauge are written as \cite{Castell,cv-sc,Gonzales,cv02}:
\beqa\label{a.9}
\Phi(\rvec,t)=\frac{1}{4\pi\eps_0}\suml_{n\ge0}\frac{(-1)^n}{n!}\,\nablav^{\,n}\vert\vert\frac{\psft^{(n)}(\tau)}{r}
\eeqa
and
\beqa\label{a.10}
\fl\;\;\;\;\;\;\;\;\;\;\;\;\Avec(\rvec,t)=\frac{\mu_0}{4\pi}\suml_{n\ge 1}\frac{(-1)^{n-1}}{n!}\left[\nablav\times\left(\nablav^{n-1}\vert\vert
\frac{\msft^{(n)}(\tau)}{r}\right)+\nablav^{n-1}\vert\vert\frac{\dot{\psft}^{(n)}(\tau)}{r}
\right]\ .
\eeqa
The dot symbolizes the time derivative and $\tau=t-r/c$ is the retarded time with respect  to the origin $O$ of the Cartesian axes in the point corresponding to the vector $\rvec$ . The origin $O$ is a point from the support of the electric charge and current  distribution. The tensors $\psft^{(n)}(t)$ and $\msft^{(n)}(t)$ are the electric and magnetic moments of the electric charges and currents
distributions $\rho(\rvec,t)$ and $\jvec(\rvec,t)$, the support of these distributions being  included in the domain
 $\dom$:
 \beqan
\psft^{(n)}(t)=\int_{\dom}\rmd^3x\;\rvec^n\,\rho(\rvec,t),
  \eeqan
  and
  \beqa\label{a.12}
\msft^{(n)}(t)=\frac{n}{n+1}\int_\dom \rmd^3x\;\rvec^n\times\jvec(\rvec,t)\ .
\eeqa
In the last equation,  a tensorial contraction via  the Levi-Civita pseudo-tensor $\eps_{ijk}$ is employed:
 \beqan
\left\{\Tsft^{(n)},\;\bbox{a}\right\}\,\to\, \Tsft^{(n)}\times \bbox{a}=\eps_{i_nqs}\Tsf_{i_1\dots i_{n-1}\,q}a_s
\esft_{i_1\dots i_n}\ ,
\eeqan
which in the particular case of $\Tsft^{(n)}=\bbox{b}^n$ becomes
\beqan
\bbox{b}^n\times\avec=b_{i_1}\,\dots\,b_{i_{n-1}}\left(\bbox{b}\times\avec\right)_{i_n}\esft_{i_1\dots i_n}\ .
\eeqan
The expansions \eref{a.9} and \eref{a.10} are running in the exterior of the minimal radius sphere including
 the support of $\rho$ and $\jvec$. \\
From equations \eref{a.9} and \eref{a.10} one obtains the following expansions of the fields $\Evec(\rvec,t)$ and $\Bvec(\rvec,t)$:
    \beqa\label{a.13}
\fl\Evec(\rvec,t)&=&-\nablav\Phi(\rvec,t)-\frac{\d\Avec(\rvec,t)}{d t}=\frac{1}{4\pi\eps_0}\suml_{n\ge 0}
\frac{(-1)^{n-1}}{n!}\nablav^{n+1}\vert\vert\frac{\psft^{(n)}(\tau)}{r}\nonumber\\
\fl&&-\frac{\mu_0}{4\pi}\suml_{n\ge 1}
\frac{(-1)^{n-1}}{n!}\left[\nablav\times\left(\nablav^{n-1}\vert\vert\frac{\dot{\msft}^{(n)}(\tau)}{r}\right)
+\nablav^{n-1}\vert\vert\frac{\ddot{\psft}^{(n)}(\tau)}{r}\right]\,
\eeqa
and
 \beqa\label{a.13a}
\fl \Bvec(\rvec,t)=\nablav\times\Avec(\rvec,t)\nonumber\\
 \fl=\frac{\mu_0}{4\pi}\suml_{n\ge 1}\frac{(-1)^{n-1}}{n!}\left[\nablav^{n+1}\vert\vert
 \frac{\msft^{(n))}(\tau)}{r}-\nablav^{n-1}\vert\vert\Delta\frac{\msft^{(n)}(\tau)}{r}+\nablav\times\left(\nablav^{n-1}\vert\vert
 \frac{\dot{\psft}^{(n)}(\tau)}{r} \right) \right].
 \eeqa

For the magnetic moments \eref{a.12}, it is also possible to introduce {\bf STF} moments
 $\mct^{(n)}=\Tcal(\msft^{(n)})$ but, this time, there are two steps required in order to complete the objective. The tensor $\msft^{(n)}$  is symmetric only in the first $n-1$ indexes    and satisfies the property
 \beqan
\msf_{i_1\dots i_{n-2}\,qq}=0\ .
\eeqan
In the first step we must obtain the symmetric projection (up to a numerical factor) $\mlrt^{(n)}$ of the tensor
$\msft^{(n)}$. We begin with the first simple example corresponding to $n=2$.  Let us write the identity
\beqan
\msf_{ij}=\frac{1}{2}\left(\msf_{ij}+ \msf_{ji}\right)+ \frac{1}{2}\left(\msf_{ij}- \msf_{ji}\right)=
\mlr_{ij}+\frac{1}{2}\eps_{ijk}\Nsf_k(\msft^{(2)})\ ,
\eeqan
 where  $\mlrt^{(2)}$ is the symmetric part of $\msft^{(2)}$ and
 \beqan \Nsf_i(\msft^{(2)})=\eps_{ijk}M_{jk}=\frac{2}{3}\int_\dom\,\rmd^3x\,\left[\rvec\times\left(\rvec\times\jvec\right)\right]\ .
 \eeqan
 In this case ($n=2$), $\mct^{(2)}=\mlrt^{(2)}$ and, consequently, corresponds to the {\bf STF} projection. Therefore,
   \beqa\label{a.15}
\msf_{ij}=\mc_{ij}+\frac{1}{2}\eps_{ijk}\Nsf_k(\msft^{(2)})\ .
 \eeqa
 For $n\ge 3$, we can generalize this result writing the identity:
 \beqa\label{a.16}
\msf_{i_1\dots i_n}&=&\frac{1}{n}\left(\msf_{i_1\dots i_n}+\msf_{i_ni_2\dots i_{n-1} i_1}+\dots+ \msf_{i_1\dots i_n i_{n-1}}
\right)\nonumber\\
&+&\frac{1}{n}\left[\left(\msf_{i_1\dots i_n}-\msf_{i_n\dots i_{n-1}i_1}\right)+\dots\left(\msf_{i_1\dots i_n}
-\msf_{i_1\dots i_{n-2}i_n\,i_{n-1}}\right)\right]\nonumber\\
&=&\mlr_{i_1\dots i_n}+\frac{1}{n}\suml^{n-1}_{\la=1}\eps_{i_\la i_nq}
\Nsf^{(\la)}_{(i_1\dots i_{n-1}q)}(\msft^{(n)})\ ,
 \eeqa
 where  by $\Nsf^{(\la)}_{(i_1\dots i_{n-1}i_n)}$ we understand the component without the index $i_\la$
 and the tensor $\Nsft^{(n-1)}=\Nsft(\msft^{(n)})$ is given by
\beqan
\fl\;\;\;\;\;\;\Nsf_{i_1\dots i_{n-1}}(\msft^{(n)})=\eps_{i_{n-1}pq}\;\msf_{i_1\dots i_{n-2}pq}=\frac{n}{n+1}\int_{\dom}
\rmd^3x\,\,x_{i_1}\dots x_{i_{n-2}}\left[\rvec\times\left(\rvec\times\jvec\right)\right]_{i_{n-1}}\ .
 \eeqan
 It is a tensor of the same type as $\msft^{(n-1)}$,  i.e.\ symmetric in the first $n-2$ indices and with
 $n-1$ vanishing  traces ($\Nsf_{i_1\dots i_{n-3}qq}=0$). Therefore, the {\bf STF} moment $\mct^{(n)}$ is given by
 the components
\beqan
\mc_{i_1\dots i_n}=\mlr_{i_1\dots i_n}-\delta_{\{i_1i_2}\lasf_{i_3\dots i_n\}}(\mlrt^{(n)})\ .
 \eeqan
As it will be seen in the dynamic case, we have to express  even the symmetric projection of the tensor $\Nsft(\msft^{(n)})$. For this,  it is useful to introduce the operator $\bbox{\mathcal N}$ defining the correspondence
\beqan
\Nsft^{(n)}\,\to\,\bbox{\mathcal N}(\Nsft^{(n)}):\;\left(\bbox{\mathcal N}(\Nsft^{(n)})\right)_{i_1\dots i_{n-1}}
=\eps_{i_{n-1}ps}\;\Nsft_{i_1\dots i_{n-2}ps}\ .
\eeqan
Repeating this operation, we obtain:
\beqan
\bbox{\mathcal N}^{2k}(\msft^{(n)})=\frac{(-1)^k\,n}{n+1}\int_{\dom}\rmd^3x\;r^{2k}\,\rvec^{n-2k}\times\jvec\ ,\nonumber\\
\bbox{\mathcal N}^{2k+1}(\msft^{(n)})=\frac{(-1)^k\,n}{n+1}\int_{\dom}\rmd^3x\;r^{2k}\,\rvec^{n-2k-1}\times(\rvec\times\jvec)\ .
\eeqan

\subsection{Delta-function identities and multipole  singularities}
 Usually, the multipole expansions are considered, term by term, as functions of $\rvec$ defined on $\mathbb{R}^3 $
 excepting a singular point which is chosen  as the origin $O$ of the Cartesian axes. Actually, these multipole terms are mathematical distributions (or generalized functions) considered firstly as regular ones having as support the entire space except the origin point $O$.  The observable quantities are expressed as weighted  averages on  spatial regions  or  as surface integrals of functions of field variables. No problems appear when in these regions  $r\ne 0$, but  when we have to calculate for example  the interaction of a distribution $(\rho,\,\jvec)$ with an external  field $(\Evec,\,\Bvec)$,
 \beqan
 W_{\scriptstyle int}=\int\rmd^3x\,\left(\rho\,\Phi-\jvec\cdot\Avec\right)
 \eeqan
 and the system associated with the external field $(\Evec,\,\Bvec)$ is represented by a point-like    multipole  system placed in $O$, the  singularities of the potentials or fields having as support this point become
 unavoidable. As done in \cite{Frahm}, these singularities are determined starting from some $\delta$-function identities associated with the extension of multiple spatial derivatives of the functions of the type $1/r$ to the entire space. Having in mind the dynamic case, too,  we generalize such identities to the derivatives of the function $f(\tau)/r$. Representing the corresponding derivatives as functions of $\rvec$ and $t$ for $r\ne0$, we can write their  expressions in the form
\beqa\label{a.19}
D_{i_1\dots i_n}(f)\equiv
\d_{i_1}\dots\d_{i_n}\frac{f(\tau)}{r}=\suml^n_{l=0}\frac{1}{c^{n-l}r^{l+1}}C^{(n,\,l)}_{i_1\dots i_n}\,\frac{\rmd^{n-l}f(\tau)}{\rmd t^{n-l}}\ ,
\eeqa
where $C^{(n,\,l)}_{i_1\dots i_n}$ are fully symmetric in the indexes $i_1\dots i_n$. The general form of the coefficients $C$ is a simple consequence of the symmetry properties and of the derivative rules:
\beqa\label{a.20}
C^{(n,\,l)}_{i_1\dots i_n}=\suml^{[\frac{n}{2}]}_{k=0}\,K^{(n,\,l)}_k\;\delta_{\{i_1i_2}\dots\delta_{i_{2k-1}i_{2k}}\,\nu_{2k+1}\dots \nu_{n\}}\ .
\eeqa
 For lower $n$ the coefficients $C$ and $K$ can be easily calculated by successive derivative operations. As one can  see in the following, for obtaining  general results for the $\delta$-type singularities of the multipole terms, it will be necessary to know only the coefficients
\beqa\label{a.22}
K^{(n,\,n)}_0=(-1)^n\,(2n-1)!!\ .
 \eeqa
In \ref{A} we enumerate  the coefficients $C^{(n\,l)}_{i_1\dots i_n}$ for the first five values of $n$.\\
Considering the distribution representing the derivative \eref{a.19} extended to the entire space, we understand by
 $\left(\d_{i_1}\dots\d_{i_n}\left(f(\tau)/r\right)\right)_{(0)}$ the singular part having as support the point $O$. In the following, we are interested  in writing explicitly  this contribution.\\
 A correct procedure for extracting this singular part is that employed in \cite{Frahm} defining
\beqa\label{a.23}
\left\langle\,\left(\d_{i_1}\dots \d_{i_n}\frac{f(\tau)}{r}\right)_{(0)},\;\phi\right\rangle=\lim_{\eps\to 0}\int_{\dom_\eps}\rmd^3x
\d_{i_1}\dots \d_{i_n}\frac{f(\tau)}{r}\,\phi(\rvec) \nonumber\\
=\lim_{\eps\to 0}\left[\oint_{\Sigma_\eps}\rmd S\,\nu_{i_1}\d_{i_2}\dots \d_{i_n}\frac{f(\tau)}{r}
-\int_{\dom_\eps}\rmd^3x\,\d_{i_2}\dots \d_{i_n}\frac{f(\tau)}{r}\,\d_{i_1}\phi(\rvec)\right]\ .
 \eeqa
 In this equation,  $\phi(\rvec)$ is supposed an element of the domain of distributions i.e.\ an  infinitely differentiable function. As it can be seen from equation \eref{a.19}, similar restrictive properties are considered for $f(\tau)$. The domain $\dom_\eps$ is the spherical region delimited  by the spherical surface $\Sigma_\eps$ with radius $\eps$. A basic hypothesis is the existence of the limits considered here, but the general demonstration of this property is a purely mathematical problem and bypasses the objectives   of the present paper.  Therefore,   the result of the limit considered is independent of the domains $\dom_\eps$ defined only by the condition $\dom_\eps\to O$.
  Finally, for the calculation process in the present paper, the essential property is the possibility to represent the functions $f$ and $\phi$ in the domain $\dom_\eps$ by their Taylor series upon the origin $O$:
 \beqa\label{a.24}
\fl f(\tau)=\suml^\infty_{\la=0}\frac{(-1)^\la\, r^\la}{c^\la\la!}\left(\frac{\rmd^\la f(\tau)}{\rmd t^\la}\right)_{r=0},\nonumber\\
\fl\phi(\rvec)=\suml^\infty_{\al=0}\frac{1}{\al!}x_{i_1}\dots x_{i_\al}\left(\d_{i_1}\dots \d_{i_\al}\,\phi(\rvec)\right)
_{\rvec=0}=\suml^\infty_{\al=0}\frac{r^\al}{\al!}\nu_{i_1}\dots \nu_{i_\al}\left(\d_{i_1}\dots \d_{i_\al}\,\phi(\rvec)\right)_{\rvec=0}\ .
 \eeqa
 Equation \eref{a.23} can be written as
 \beqan
 \fl\left\langle\,\left(D_{i_1\dots i_n}(f)\right)_{(0)},\;\phi\right\rangle=\lim_{\eps\to 0}\oint_{\Sigma_\eps}\rmd S\,\nu_{i_1}\d_{i_2}\dots \d_{i_n}\frac{f(\tau)}{r}-\left\langle\,\left(D_{i_2\dots i_n}(f)\right)_{(0)},\;\d_{i_1}\phi\right\rangle
 \eeqan
 generating  a recursive calculation for a given $n$. \\
Let us express the $\delta$-singularities for lower order derivatives. Even in these simple cases, it becomes obvious the importance of  the angular average of a function $g(\nuvec)$ defined as
\beqan
\langle g(\nuvec)\rangle=\frac{1}{4\pi}\int\,g(\nuvec)\,\rmd\Omega(\nuvec)\ .
\eeqan
For this average, we have the  well known    formula \cite{Thorne}:
\beqa\label{3.6}
\langle\nu_{i_1}\dots\nu_{i_n}\rangle=\left\{\begin{array}{c}0,
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n=2k+1,\\
\frac{1}{(n+1)!!}\,\delta_{\{i_1i_2}\dots\delta_{i_{n-1}i_n\}},
\;\;\;\;\;\;\;\;\;\;n=2k,\;\;\;\;\;k=0,1,\dots\end{array}\right.
\eeqa
The first $\delta$-singularity is obtaining for $n=2$ calculating the limit
\beqan
\fl\left\langle\,\left(D_{ij}(f)\right)_{(0)},\;\phi\right\rangle=\lim_{\eps\to0}\left[\oint_{\Sigma_\eps}\rmd S\,\nu_i\left(\d_j\frac{f(\tau)}{r}\right)\,\phi(\rvec)-\int_{\dom_\eps}\rmd^3x\,\left(\d_j\frac{f(\tau)}{r}\right)\,\d_i\phi(\rvec)\right]\\
\fl=\lim_{\eps\to0}\left[\oint_{\Sigma_\eps}\rmd S\,\nu_i\left(\d_j\frac{f(\tau)}{r}\right)\,\phi(\rvec)-\oint_{\Sigma_\eps}\frac{\rmd S}{r}\,\nu_j\,f(\tau)\,\d_i\phi(\rvec)+\int_{\dom_\eps}\frac{\rmd^3x}{r}\,f(\tau)\,\d_i\d_j\phi(\rvec)
\right]\ .
\eeqan
The first surface integral limit can be written as
\beqan
\bbox{L}_{1\sigma}=\lim_{\eps\to0}\oint\rmd\Omega(\nuvec)\;\eps^2\,\nu_i\left[-\frac{\nu_j}{c\eps} \dot{f}(\tau_\eps)
-\frac{\nu_j}{\eps^2}f(\tau_\eps)\right]\,\left[\phi(0)+\eps\nu_k\left(\d_k\phi\right)_0+\dots\right],
\eeqan
where $\tau_\eps=t-\eps/c$. Since $f(\tau_\eps)=f(t)+\mathcal{O}(\eps)$, and all the terms containing as factors positive powers of $\eps$ vanish at the limit $\eps\to0$, we can write
\beqan
\bbox{L}_{1\sigma}=-4\pi\left\langle\nu_i\nu_j\right\rangle\,f(t)\phi(0)=-\frac{4\pi}{3}\delta_{ij}\;f(t)\phi(0)\ .
\eeqan
The second surface integral limit
\beqan
\bbox{L}_{2\sigma}=\lim_{\eps\to0}\oint\rmd\Omega(\nuvec)\,\eps\nu_j\,f(\tau_\eps)\left[\left(\d_i\phi\right)_0+\eps\nu_k\left(\d_i\d_k\phi\right)_0+\dots\right]\,=\,0\ ,
\eeqan
since all the terms contain as factors positive powers of $\eps$. As the  volume integral limit also vanishes,  we can  finally write:
 \beqa\label{3.5a}
\left(\d_i\d_j\frac{f(\tau)}{r}\right)_{(0)}=-\frac{4\pi}{3}f(t)\delta_{ij}\,\delta(\rvec)\ .
\eeqa
Including the expression of the derivative for $r\ne 0$, the above result written for $f(t)=1$ verifies equation \eref{1.3}.\\
Let us consider the next order derivative and calculate
\beqan
\fl\left\langle\,\left(D_{ijk}(f)\right)_{(0)},\,\phi\right\rangle=\lim_{\eps\to0}\left[\oint_{\Sigma_\eps}\rmd S
\nu_i\left(\d_j\d_k\frac{f(\tau)}{r}\right)\,\phi(\rvec)-\int_{\dom_\eps}\rmd^3x\left(\d_j\d_k\frac{f(\tau)}{r}\right)\d_i\phi(\rvec)\right]=\\
\fl\lim_{\eps\to0}\left\{\oint\rmd\Omega(\nuvec)\,\eps^2\left[\frac{1}{c^2\eps}\nu_i\nu_j\nu_k\ddot{f}(\tau_\eps)+\frac{1}{c\eps^2}\nu_i(3\nu_j\nu_k-\delta_{jk})\dot{f}(\tau_\eps)
+\frac{1}{\eps^3}\nu_i(3\nu_j\nu_k-\delta_{jk})f(\tau_\eps)\right]\right.\\
\left. \times
\left(\phi(0)+\eps\nu_l\left(\d_l\phi\right)_0+\dots\right).
+\frac{4\pi}{3}\int_{\dom_\eps}\rmd^3x\,f(t)\delta_{jk}\delta(\rvec)\,\d_i\phi(\rvec)\right\}\ ,
\eeqan
where equation \eref{3.5a} is employed in the volume integral. In the surface integral limit, as $\eps\to 0$,  the   terms containing as factor the second order time derivative $\ddot{f}$ vanish because of the positive powers of $\eps$. The terms containing the first order time derivative $\dot{f}$ are proportional to  positive powers of $\eps$  for the term corresponding to $\phi(0)$, but in this case the corresponding limit cancels  since it contains also as factors the null averages $\langle\nu_i\nu_j\nu_k\rangle$ and $\langle\nu_i\rangle$. The only terms giving a non vanishing limit are provided by the product $f(\tau_\eps)\,\left(\d_l\phi\right)_0$ such that we can write
\beqan
\fl\left\langle\,\left(D_{ijk}(f)\right)_{(0)},\,\phi\right\rangle &=&4\pi\left\langle3\nu_i\nu_j\nu_k\nu_l-\delta_{jk}\nu_i\nu_l\right\rangle\,f(t)\left(\d_l\phi\right)_0+\frac{4\pi}{3}f(t)\delta_{jk}\left(\d_i\phi\right)_0\\
&=&4\pi\left(\frac{1}{5}\delta_{\{ij}\delta_{kl\}}-\frac{1}{3}\delta_{jk}\delta_{il}\right)f(t)\left(\d_l\phi\right)_0
+\frac{4\pi}{3}f(t)\delta_{jk}\left(\d_l\phi\right)_0\ .
\eeqan
Finally:
\beqa\label{3.5b}
\left(\d_i\d_j\d_k\frac{f(\tau)}{r}\right)_{(0)}=-\frac{4\pi}{5}f(t)\delta_{\{ij}\d_{k\}}\delta(\rvec)\ .
\eeqa
For $f(t)=1$, equation \eref{3.5b} becomes the $\delta$-singularity corresponding to equation (13) from Ref.\ \cite{Frahm}. Equations \eref{3.5a}  and \eref{3.5b} will be applied in the static case for $f(t)=1$.

\section{Singularities of the electrostatic field}\label{estatic}
\par Let us consider the multipole expansion of the electrostatic field derived from the potential expansion \eref{a.9} in the static case:
 \beqan
\fl\;\;\;\;\;\Evec(\rvec)=\frac{1}{4\pi\eps_0}\suml_{n\ge 0}\frac{(-1)^{n-1}}{n!}\nablav^{n+1}\vert\vert\frac{\psft^{(n)}}{r}=
\frac{1}{4\pi\eps_0}\suml_{n\ge 0}\frac{(-1)^{n-1}}{n!}\psft^{(n)}\vert\vert\nablav^{n+1}\frac{1}{r}\ .
\eeqan
In the dipolar case ($n=1$), the potential has no $\delta$-singularities. For the electric field, we  apply equation \eref{3.5a}  obtaining the known result \eref{1.1}.\\
A first interesting feature of the $\delta$-singularities problem appears beginning with the electric quadrupole  potential and field. The singularity of the potential is obtained employing  equation \eref{3.5a}:
\beqan
\Phi^{(2)}(\rvec)=\left(\frac{1}{8\pi\eps_0}\nablav^2\vert\vert\frac{\psft^{(2)}}{r}\right)_{r\ne 0}-\frac{1}{6\eps_0}\psf_{qq}\delta(\rvec)\ .
\eeqan
The first term from the right-hand side of this equation is invariant to the substitution $\psft^{(2)}\,\to\,\pct^{(2)}=\Tcal(\psft^{(2)})$  and, employing equations \eref{a.2} and \eref{a.3} with $ \Ssft=\psft^{(2)}$, we can write
\beqan
\Phi^{(2)}(\rvec)
=\left(\frac{1}{8\pi\eps_0}\nablav^2\vert\vert\frac{\pct^{(2)}}{r}\right)_{r\ne0}
-\frac{1}{2\eps_0}\lasf\;\delta(\rvec)\ .
\eeqan
Usually, since  in the exterior of the charge distribution the potential $\Phi(\rvec)$ is solution of the Laplace equation, it is represented by a series of spherical functions:
\beqa\label{3.9}
\Phi(\rvec)=\suml^\infty_{l=0}\frac{1}{r^{l+1}}\suml^l_{m=-l}Q_{lm}\,Y_{lm}(\theta,\varphi)\ .
\eeqa
The $2l+1$ spherical moments corresponding to a given $l$ are linear combinations of the $2l+1$ components of the $l$-th order {\bf STF} tensor $\pct^{(l)}$. For the above example of the 4-polar potential, it is obvious that using
 the expansion representing the particular case of the equation \eref{3.9}, the $delta$-singularity of the potential is lost. The conclusion can be extended to the higher-order multipolar potentials. For the electrostatic field $\Evec(\rvec)$, the employment of the corresponding  expansion in spherical functions  leads to an incomplete description of the $\delta$-form singularities.\\
 Based on this observation, in our opinion, some care is necessary when passing to the representations by spherical functions since it is equivalent to working only in a subspace of the  mathematical objects used correctly in the theory. The  direct substitution $\psft\,\to\,\pct$ or passing to the spherical tensorial representations is not always justified. This    observation is pointed out  in another context in \cite{Raab}, too.\\
Searching the $\delta$-singularities of $\Evec^{(2)}(\rvec)$, instead of a straightforward processing of the corresponding expression in terms of the primitive moment $\psft^{(2)}$, we can introduce from the beginning the {\bf STF} moment $\pct^{(2)}$ writing
\beqan
\fl\;\;\;\;\;\;\;\;\;\left(\Evec^{(2)}(\rvec)\right)_{(0)}&=&-\frac{1}{8\pi\eps_0}\left(\nablav^3\vert\vert\frac{\psft^{(2)}}{r}
\right)_{(0)}=-\frac{1}{8\pi\eps_0}\left(\nablav^3\vert\vert\frac{\pct^{(2)}}{r}+\evec_i\d_i\Delta\frac{\lasf}{r}
\right)_{(0)}\nonumber\\
\fl\;\;\;\;\;\;\;\;\;&=&-\frac{1}{8\pi\eps_0}\left(\nablav^3\vert\vert\frac{\pct^{(2)}}{r}\right)_{(0)}+\frac{1}{2\eps_0}\lasf\delta(\rvec)\ ,
\eeqan
since $\Delta(1/r)=-4\pi\delta(\rvec)$. Equation \eref{3.5b}  finally gives
\beqa\label{3.10}
\fl\;\;\;\;\;\;\;\;\;\Evec^{(2)}(\rvec)=\left(\Evec^{(2)}(\rvec)\right)_{r\ne0}+\frac{1}{5\eps_0}\pct^{(2)}\vert\vert\nablav\delta(\rvec)+\frac{1}{2\eps_0}\lasf\nablav\delta(\rvec),\;\;\;\lasf=\latens(\psft^{(2)})\ ,
\eeqa
where
\beqan
\left(\Evec^{(2)}(\rvec)\right)_{r\ne0}=-\frac{1}{8\pi\eps_0}\left(\nablav^3\vert\vert\frac{\psft^{(2)}}{r}\right)_{r\ne0}=-\frac{1}{8\pi\eps_0}\left(\nablav^3\vert\vert\frac{\pct^{(2)}}{r}\right)_{r\ne0}\ .
\eeqan
Even from this simple example it becomes obvious that this calculation version is convenient  for higher $n$ since some contractions of $\pct^{(n)}$ with the Kronecker symbols vanish. \\
In  case $n=3$, the advantage of inserting from the very beginning the {\bf STF} moments is more evident. Instead of searching the singularities of the fourth-order derivative $\d_i\d_j\d_k\d_l\,(1/r)$, we search the singularities of the contraction of this derivative with the {\bf STF} moment $\pct^{(3)}$ writing
\beqan
\fl\left\langle\left(\nablav^4\vert\vert\frac{\pct^{(3)}}{r}\right)_{(0)},\phi\right\rangle
=\evec_i\lim_{\eps_0\to0}\left[\oint_{\Sigma_\eps}\rmd S\,\nu_i\d_j\d_k\d_l\frac{\pc_{jkl}}{r}\,\phi(\rvec)
-\int_{\dom_\eps}\rmd^3x\,\d_j\d_k\d_l\frac{\pc_{jkl}}{r}\,\d_i\phi(\rvec)\right].
\eeqan
For the surface integral limit, we apply equation \eref{a.19} particularized to the static case. Denoting by $\bbox{L}_\sigma$ this limit,
\beqan
\bbox{L}_{\sigma}=\evec_i\lim_{\eps\to 0}\oint_{\Sigma_\eps}\frac{\rmd\Omega(\nuvec)}{\eps^2}\nu_i\,C^{(3,\,3)}_{jkl}\pc_{jkl}\,\phi(\rvec)
\eeqan
and, inserting equation \eref{C3}, one obtains
\beqan
\bbox{L}_{\sigma}&=&\evec_i\lim_{\eps\to 0}\oint_{\Sigma_\eps}\frac{\rmd\Omega(\nuvec)}{\eps^2}\left(-15\nu_i\nu_j\nu_k\nu_l+3\nu_i\delta_{\{jk}\nu_{l\}}\right)\pc_{jkl}\,\phi(\rvec)\ .
\eeqan
Since $\delta_{\{jk}\nu_{l\}}\pc_{jkl}=0$, we get
\beqan
\fl\;\;\;\bbox{L}_{\sigma}&=&-15\evec_i\lim_{\eps\to 0}\oint_{\Sigma_\eps}\frac{\rmd\Omega(\nuvec)}{\eps^2}\nu_i\nu_j\nu_k\nu_l
\pc_{jkl}\left(\phi(0)+\eps\nu_q\left(\d_q\phi\right)_0+\frac{1}{2}\eps^2\nu_q\nu_s\left(\d_q\d_s\phi\right)_0\right)\\
\fl&=&-4\pi\times 15\,\evec_i\lim_{\eps\to 0}\left\langle\,\frac{1}{\eps^2}\nu_i\nu_j\nu_k\nu_l\,\pc_{jkl}
\left(\phi(0)+\eps\nu_q\left(\d_q\phi\right)_0+\frac{1}{2}\eps^2\nu_q\nu_s\left(\d_q\d_s\phi\right)_0\right)\right\rangle\ .
\eeqan
Denoting by $\al$ the power of $\eps$ in the Taylor series of $\phi(\rvec)$, we easily see  that for $\al\,\ge\,3$ all
the terms from the last equation vanish for $\eps\,\to\,0$. For $\al=0$, the corresponding term vanishes since $\langle\nu_i\nu_j\nu_k\nu_l\rangle\,\pc_{jkl}=0$,  as it is seen by inserting equation \eref{3.6}. For $\al=1$, the related  term contains the average of an odd number of factors $\nu$ yielding a null result, too. The only non vanishing result corresponds to $\al=2$ such that
\beqan
\bbox{L}_\sigma=-4\pi\times\frac{15}{2}\,\evec_i\left\langle\nu_i\nu_j\nu_k\nu_l\nu_q\nu_s\right\rangle\,\pc_{jkl}\ .
\eeqan
In the last equation we have the contraction  $\langle\nu_{i_1}\nu_{i_2}\nu_{i_3}\nu_{i_4}\nu_{i_5}\nu_{i_6}\rangle\pc_{i_4i_5i_6}$. It is not necessary to write explicitly the angular average of the six factors $\nu$ since, considering the general form of this  expression given in equation \eref{3.6}, we see that only the $3!=6$ terms of the form $\delta_{i_1i_j}\delta_{i_2i_k}\delta_{i_3i_l}$ with the  indexes $j,k,l$ covering  all the permutations of $(4,5,6)$ give non vanishing limits. Consequently,
\beqa\label{3.10a}
\bbox{L}_\sigma=-4\pi\frac{15\times3!}{2\times 7!!}\evec_i\pc_{ijk}\left(\d_j\d_k\phi\right)_0
=-\frac{24\pi}{14}\pct^{(3)}\vert\vert\left(\nablav^2\phi\right)_0\ .
\eeqa
The volume integral limit is zero since
\beqan
\left(\nablav^3\vert\vert\frac{\pct^{(3)}}{r}\right)_{(0)}=0\ ,
\eeqan
as it can be easily verified. Equation \eref{3.10a} implies
\beqa\label{3.10bb}
\left(\nablav^4\vert\vert\frac{\pct^{(3)}}{r}\right)_{(0)}=-\frac{24}{14}\;\pct^{(3)}\vert\vert\nablav^2\delta(\rvec)\ .
\eeqa
 In  \ref{B} this equality is proven for the general case.
Let us write
\beqan
\fl\;\;\;\;\;\left(\Evec^{(3)}(\rvec)\right)_{(0)}=\frac{1}{24\pi\eps_0}\left(\nablav^4\vert\vert\frac{\psft^{(3)}}{r}\right)_{(0)}
=-\frac{1}{14\eps_0}\pct^{(3)}\vert\vert\nablav^2\delta(\rvec)-\frac{1}{2\eps_0}\bbox{\Lambda}\vert\vert\nabla^2\delta(\rvec)
\eeqan
or
\beqan
\Evec^{(3)}(\rvec)=\left(\Evec^{(3)}(\rvec)\right)_{r\ne 0}-\frac{1}{14\eps_0}\pct^{(3)}\vert\vert\nablav^2\delta(\rvec)-\frac{1}{2\eps_0}\bbox{\Lambda}\vert\vert\nabla^2\delta(\rvec)\ ,
\eeqan
where
\beqan
\left(\Evec^{(3)}(\rvec)\right)_{r\ne 0}=\frac{1}{24\pi\eps_0}\nablav^4\vert\vert\frac{\psft^{(3)}}{r}=
\frac{1}{24\pi\eps_0}\nablav^4\vert\vert\frac{\pct^{(3)}}{r}\ .
\eeqan
Even  if only for the  mathematical interest, we mention the possibility of expressing the multipole electrostatic field for arbitrary $n$. In \ref{B} it is introduced the general relation corresponding to equation \eref{3.10bb}:
\beqan
\left(\nablav^{n+1}\vert\vert\frac{\pct^{(n)}}{r}\right)_{(0)}=-\frac{4\pi\,n}{2n+1}\,\pct^{(n)}\vert\vert\nablav^{n-1}\delta(\rvec)
\eeqan
such that
\beqa\label{3.14}
\fl\;\;\;\;\;\;\left(\Evec^{(n)}(\rvec)\right)_{(0)}&=&\frac{(-1)^{n-1}}{4\pi\eps_0\,n!}\left(\nablav^{n+1}\vert\vert\frac{\pct^{(n)}}{r}+\frac{n(n-1)}{2}\nablav^{n-1}\vert\vert\Delta\frac{\latens^{(n-2)}}{r}\right)_{(0)}\nonumber\\
\fl\;\;\;\;\;\;&=&\frac{(-1)^n\,n}{\eps_0}\left(\frac{1}{n!\,(2n+1)}\pct^{(n)}+\frac{n-1}{2n!}\latens^{(n-2)}\right)\vert\vert\nablav^{n-1}\delta(\rvec)\ .
\eeqa
We point out that for $n\ge 4$,  for obtaining the complete  singular structure of the field,  we have to introduce  the irreducible tensors in the expression from equation \eref{3.14} corresponding to the $\delta$-singularities .
This implies a recursive calculation introducing in equation \eref{3.14} the {\bf STF} projections of $\latens^{(n-2)},\,\latens^{(n-4)}, \dots$. For example, in the case $n=4$, we have to reduce the tensor $\latens^{(2)}=\latens(\psft^{(4)})$ writing:
\beqan
\latens^{(2)}\vert\vert\nablav^3\delta(\rvec)=\Tcal(\latens^{(2)})\vert\vert\nablav^3\delta(\rvec)+\latens(\latens^{(2)})\,\nablav\Delta\delta(\rvec)\ ,
\eeqan
where
\beqan
\lasf(\latens^{(2)})=\frac{1}{30}\psf_{qqss}\ ,\\
\mathcal{T}_{ij}(\latens^{(2)})=\frac{1}{7}\left(\psf_{qqij}-\frac{1}{3}\psf_{qqss}\delta_{ij}\right)\ .
\eeqan

\section{Singularities of the magnetostatic field}\label{magnetost}
\par Let us write the multipole expansions of the vector potential $\Avec(\rvec)$ and of the  magnetic field $\Bvec(\rvec)$ in the static case:
\beqan
\Avec(\rvec)&=&\frac{\mu_0}{4\pi}\suml_{n\ge 1}\frac{(-1)^{n-1}}{n!}\nablav\times\left(\nablav^{n-1}\vert\vert\msft^{(n)}\right)
\eeqan
and, respectively,
\beqa\label{4.3}
\Bvec(\rvec)=
\frac{\mu_0}{4\pi}\suml_{n\ge 1}\frac{(-1)^{n-1}}{n!}\left[\nablav\left(\nablav^n\vert\vert\frac{\msft^{n)}}{r}\right)
-\Delta\left(\nablav^{n-1}\vert\vert\frac{\msft^{(n)}}{r}\right)\right]\ .
 \eeqa
 For $r\ne 0$, the multipole magnetic field $\Bvec(\rvec)$ is expressed, as in the electric case, only by a scalar potential since the second term from the right-hand side of equation \eref{4.3} vanishes. However, the $\delta$-singularity of the magnetic field is not similar to the corresponding singularity of the electric field,   this second term giving the difference. The difference becomes easily obvious if one considers the fictitious magnetic shells (or sheets) employed in the Amp\`{e}re formalism.
 It suffices to consider the case of the point-like magnetic dipole which can be taken as the limit of a current loop of infinitesimal size. For finite dimensions, the field of the loop  is derived from a scalar potential $\Phi_m$
 which is defined by an integral on the corresponding sheet and having a ``jump''  in all their points. Just this jump generates the  $\delta$-singularity corresponding to the second term from equation \eref{4.3}. An explicit calculation of this limit when the loop concentrates in a point is given in Ref.\ \cite{Corbo}.\\
 In the dipolar case ($n=1$), retaining the singular term, we have
 \beqan
 \left(\Bvec^{(1)}(\rvec)\right)_{(0)}=\frac{\mu_0}{4\pi}\left(\evec_i\d_i\d_j\frac{m_j}{r}\right)_{(0)}+4\pi\,\mvec\delta(\rvec)\ .
 \eeqan
 Processing the first term as in the electrostatic case, we obtain  the well-known expression \eref{1.2}.
No $\delta$-singularity is present in the dipolar vector potential.\\
Let us consider the quadrupole term from equation \eref{4.3} and the corresponding singularities:
\beqan
\left(\Bvec^{(2)}(\rvec)\right)_{(0)}=-\frac{\mu_0}{8\pi}\left(\nablav^3\vert\vert\frac{\msft^{(2)}}{r}-\nablav\vert\vert\left(\Delta\frac{\msft^{(2)}}{r}\right)\right)_{(0)}\ .
\eeqan
By introducing the {\bf STF} moment $\mc^{(2)}=\mlrt^{(2)}$ defined by equation \eref{a.15},
\beqan
\fl\left(\Bvec^{(2)}(\rvec)\right)_{(0)}&=&-\frac{\mu_0}{8\pi}\left(\nablav^3\vert\vert\frac{\mct^{(2)}}{r}
+\frac{1}{2}\evec_i\eps_{jkl}\d_i\d_j\d_k\frac{\Nsf_k}{r}-\evec_i\d_j\Delta\frac{\mc_{ji}}{r}
+\frac{1}{2}\eps_{ijl}\d_j\Delta\frac{\Nsf_l}{r}\right)_{(0)}\\
\fl &=&-\frac{3\mu_0}{10}\mct^{(2)}\vert\vert\nablav\delta(\rvec)-\frac{\mu_0}{4}\bbox{N}\times\nablav\delta(\rvec)\ ,
\eeqan
where $\bbox{N}=\evec_i\Nsf_i$. It follows that
\beqa\label{4.6}
\Bvec^{(2)}(\rvec)=\left(\Bvec^{(2)}(\rvec)\right)_{r\ne0}
-\frac{3\mu_0}{10}\,\mct^{(2)}\vert\vert\nablav\delta(\rvec)-\frac{\mu_0}{4}\,\bbox{N}\times\nablav\delta(\rvec)
\eeqa
where
\beqan
\left(\Bvec^{(2)}(\rvec)\right)_{r\ne0}=\left(-\frac{\mu_0}{8\pi}\nablav^3\vert\vert\frac{\mct^{(2)}}{r}\right)_{r\ne0}=\left(-\frac{\mu_0}{8\pi}\nablav^3\vert\vert\frac{\msft^{(2)}}{r}\right)_{r\ne0}\ .
\eeqan

Let us consider the quadrupolar vector potential $\Avec^{(2)}(\rvec)$ with equation \eref{a.15} already inserted:
 \beqa\label{4.8}
\Avec^{(2)}(\rvec)&=&-\frac{\mu_0}{8\pi}\evec_i\eps_{ijk}\,\d_j\d_l\frac{\msf_{lk}}{r}
=-\frac{\mu_0}{8\pi}\nablav\times\left(\nablav\vert\vert\frac{\mct^{(2)}}{r}\right)\nonumber\\
&&+\frac{\mu_0}{16\pi}\nablav\left(\nablav\cdot\frac{\bbox{N}}{r}\right)
-\frac{\mu_0}{16\pi}\bbox{N}\Delta\frac{1}{r}\ .
  \eeqa
 The term proportional to $\Delta(1/r)$ vanishes   for $r\ne 0$. Whereas $\Bvec^{(2)}$ is invariant to the substitution $\psft^{(2)}\to \pct^{(2)}$, this property is not verified by the vector potential $\Avec^{(2)}$.
 This substitution has as result an additional gradient term, i.e.\ a gauge transformation of the potential.
Though, by the extension to the entire space, the gradient term  from expression \eref{4.8} generates  a $\delta$-singularity, this term gives no contribution to the singularities of the magnetic field The singularities of $\Bvec$  are, by definition, only the singularities corresponding to the curl of the vector potential. It remains still the problem of  independence
of the physical results on the $\Avec$ $\delta$-singularities corresponding to different gauges. Maybe, the presence of such singularities in the expression of an interaction Hamiltonian corresponding to the density term $ \jvec\cdot\Avec$ can generate the problem of proving the gauge independence of physical results as in the case of the Aharonov-Bohm effect.\\
We can give, as in the electrostatic case, the general formula for the singularities of the magnetic field writing
 \beqa\label{4.10}
\left(\Bvec^{(n)}(\rvec\right)_{(0)}=\frac{(-1)^{n-1}\mu_0}{4\pi \,n!}\left(\nablav^{n+1}\vert\vert\frac{\msft^{(n)}}{r}-
\nablav^{n-1}\vert\vert\Delta\frac{\msft^{(n)}}{r}\right)_{(0)}\ .
\eeqa
The introduction of the symmetric moment $\mlrt^{(n)}$, equation \eref{a.16}, gives
\beqan
\fl\;\;\;\;\;\;\nablav^{n+1}\vert\vert\frac{\msft^{(n)}}{r}=
\nablav^{n+1}\vert\vert\frac{\mlrt^{(n)}}{r}
+\frac{1}{n}\evec_i\d_i\d_{i_1}\dots\d_{i_n}\suml^{n-1}_{\la=1}\eps_{i_\la i_n q}
\frac{\Nsf^{(\la)}_{(i_1\dots  i_{n-1}q)}}{r}=\nablav^{n+1}\vert\vert\frac{\mlrt^{(n)}}{r}\ ,
\eeqan
since in all terms of the sum there are the contractions $\eps_{i_li_nq}\d_{i_l}\d_{i_n}$ for $l=1,\dots n-1$ which vanish. From equation \eref{3.14} we can write a corresponding result for the magnetic field with the substitutions $\psft^{(n)}\to \mlrt^{(n)},\;\pct^{(n)}\to\mct^{(n)},\;\eps_0\to 1/\mu_0$:
\beqan
\fl\;\;\;\;\;\;\frac{(-1)^{n-1}\mu_0}{4\pi\,n!}\left(\nablav^{n+1}\vert\vert\frac{\mlrt^{(n)}}{r}\right)_{(0)}&=&
(-1)^n\mu_0\left(\frac{n}{n!\,(2n+1)}\mct^{(n)}\vert\vert\nablav^{n-1}\delta(\rvec)\right.\nonumber\\
\fl\;\;\;\;\;\;&&+\left.\frac{n(n-1)}{2\,n!}\widetilde{\latens}^{(n-2)}\vert\vert\nablav^{n-1}\delta(\rvec)\right)\ ,
\eeqan
where $\widetilde{\latens}^{(n-2)}=\latens(\mlrt^{(n)})$.
The second term from equation \eref{4.10} is processed as follows:
\beqan
\nablav^{n-1}\vert\vert\Delta\frac{\msft^{(n)}}{r}&=\nablav^{n-1}\vert\vert\Delta\frac{\mlrt^{(n)}}{r}
+\frac{1}{n}\evec_i\d_{i_1}\dots\d_{i_{n-1}}\suml^{n-1}_{\la=1}\eps_{i_\la iq}\Delta\frac{\Nsf^{(\la)}_{(i_1\dots i_{n-1}q)}}{r}\\
&=\nablav^{n-1}\vert\vert\Delta\frac{\mlrt^{(n)}}{r}+\frac{n-1}{n}\evec_i\,\eps_{iqi_{n-1}}\,\Nsf_{i_1\dots i_{n-2}q}\d_{i_1}\dots\d_{i_{n-1}}\Delta\frac{1}{r}\ .
\eeqan
 The term containing $\mlrt$ can be further transformed to:
\beqan
\fl&&\mlrt^{(n)}\vert\vert\nablav^{n-1}\delta(\rvec)=\mct^{(n)}\vert\vert\nablav^{n-1}\delta(\rvec)+
\evec_i\,\d_{i_1}\dots\d_{i_{n-1}}\,\delta(\rvec)\delta_{\{i_1i_2}\widetilde{\lasf}_{i_3\dots i_{n-1}\,i\}}\\
\fl&=&\mct^{(n)}\vert\vert\nablav^{n-1}\delta(\rvec)+\frac{(n-1)(n-2)}{2}\evec_i\d_{i_1}\dots\d_{i_{n-3}}\Delta\delta(\rvec)\widetilde{\lasf}_{i_1\dots i_{n-3}i}\\
\fl&&+(n-1)\evec_i\d_i\,\d_{i_1}\dots \d_{i_{n-2}}\delta(\rvec)\widetilde{\lasf}_{i_1\dots i_{n-2}}\\
\fl&=&\mct^{(n)}\vert\vert\nablav^{n-1}\delta(\rvec)+\frac{(n-1)(n-2)}{2}\widetilde{\latens}^{n-2}\vert\vert\nablav^{n-3}\Delta\delta(\rvec)+(n-1)\widetilde{\latens}^{n-2}\vert\vert\nablav^{n-1}\delta(\rvec)\ .
\eeqan
Finally,
\beqa\label{finmst}
\fl\;\;\;\;\;\left(\Bvec^{(n)}(\rvec)\right)_{(0)}&=&\frac{(-1)^n\mu_0}{n!}\left[\left(-\frac{n+1}{2n+1}\mct^{(n)}+\frac{(n-1)(n-2)}{2}\widetilde{\latens}^{(n-2)}\right)\vert\vert\nablav^{n-1}\delta(\rvec)\right.\nonumber\\
\fl\;\;\;\;\;&-&\left.\frac{n-1}{n}\Nsft^{(n-1)}\vert\times\vert\nablav^{n-1}\delta(\rvec)
-\frac{(n-1)(n-2)}{2}\widetilde{\latens}^{(n-2)}\vert\vert\nablav^{n-3}\Delta\delta(\rvec)
\right]
\eeqa
where  the notation
\beqan
\left(\Asft^{(n)}\vert\times\vert\Bsft^{(n)} \right)_{i_1\dots i_n}=\eps_{i_nqs}\Asf_{i_1\dots i_{n-1}\,q}\Bsf_{i_1\dots i_{n-1}\,s}
\eeqan
is introduced. The general expression is written  as
\beqan
\Bvec^{(n)}(\rvec)=\left(\Bvec^{(n)}(\rvec)\right)_{r\ne 0}+\left(\Bvec^{(n)}(\rvec)\right)_{(0)}(\rvec)\ ,
\eeqan
where
\beqan
\left(\Bvec^{(n)}(\rvec)\right)_{r\ne 0}=\frac{(-1)^{n-1}\mu_0}{4\pi\,n!}\nablav^{n+1}\vert\vert\frac{\mct^{(n)}}{r}
=\frac{(-1)^{n-1}\mu_0}{4\pi\,n!}\nablav^{n+1}\vert\vert\frac{\msft^{(n)}}{r}\ ,
\eeqan
with the derivatives calculated for $r\ne 0$.\\
Here is the place for the  same observation as at the end of Section \eref{estatic}, but in the singular term containing $\Nsft^{(n-1)}$
the reduction to the {\bf STF} tensor $\Tsft^{(n-1)}$ begins from $n=3$. Particularly,
\beqan
 \Tcal(\Nsft^{(2)})=\stackrel{<->}{\Nsft}^{(2)},\;\;\;  \stackrel{<->}{\Nsf}_{ij}=\frac{1}{2}\left(\Nsf_{ij}+\Nsf_{ji}\right)\ ,
\eeqan
and
\beqan
\fl\;\;\;\;\;\;\Nsft^{(2)}\vert\times\vert\nablav^2\delta(\rvec)=\stackrel{<->}{\Nsft}^{(2)}\vert\times\vert\nablav^2\delta(\rvec)
+\frac{1}{2}\Nmc(\Nsft^{(2)})\vert\vert\nablav^2\delta(\rvec)-\frac{1}{2}\Nmc(\Nsft^{(2)})\Delta\delta(\rvec)\ .
\eeqan
Obviously, the analysis of the  higher order singular terms    is more complicated for the magnetic field  than for the  electric one,  but it is  purely a technical problem.

\section{The dynamic case}\label{dynamic}
Considering the $n$-th order multipole  electric and magnetic fields, writing separately the  contributions of  the electric and magnetic multipoles,

 \beqa\label{5.1}
 \Evec^{(n,\,p)}(\rvec,t)=\frac{(-1)^{n-1}}{4\pi\eps_0\,n!}\left(\nablav^{n+1}\vert\vert\frac{\psft^{(n)}(\tau)}{r}
-\frac{1}{c^2}\nablav^{n-1}\vert\vert\frac{\ddot{\psft}^{(n)}(\tau)}{r}\right),\nonumber\\
\Evec^{(n,\,m)}(\rvec,t)=\frac{(-1)^n}{4\pi\eps_0c^2\,n!}\nablav\times\left(\nablav^{n-1}\vert\vert\frac{\dot{\msft}^{(n)}(\tau)}{r}\right)
\eeqa
 and
  \beqa\label{5.1a}
 \Bvec^{(n,\,p)}(\rvec,t)=\frac{(-1)^{n-1}\mu_0}{4\pi\,n!}\nablav\times\left(\nablav^{n-1}\vert\vert\frac{\dot{\psft}^{(n)}(\tau}{r}\right)\ ,\nonumber\\ \Bvec^{(n,\,m)}(\rvec,t)=\frac{(-1)^{n-1}\mu_0}{4\pi\,n!}\left(\nablav^{n+1}\vert\vert\frac{\msft^{(n)}(\tau)}{r}-\nablav^{n-1}\vert\vert\Delta\frac{\msft^{(n)}(\tau)}{r}
\right)\ ,
 \eeqa
we search the corresponding $\delta$-singularities. Processing  the different   terms from equation \eref{5.1} and \eref{5.1a} for reducing the moment tensors to the {\bf STF} ones, we deal with the Laplace operator $\Delta$ applied to functions of the type $f(\tau)/r$. Since the equation verified by these functions is
 \beqa\label{5.2}
\Delta\frac{f(\tau)}{r}-\frac{1}{c^2}\frac{\d^2}{\d t^2}\frac{f(\tau)}{r}=-4\pi f(t)\delta(\rvec)\ ,
\eeqa
the processing is more complicated in the dynamic case compared with the static one. \\
 For the electric field of the electric dipole we write the singular part as
 \beqan
\fl\left(\Evec^{(1,\,p)}(\rvec,t)\right)_{(0)}=\frac{1}{4\pi\eps_0}\left[\nablav\left(\nablav\cdot\frac{\pvec(\tau)}{r}\right)-\frac{1}{c^2}\frac{\ddot{\pvec}(\tau)}{r}\right]_{(0)}=\frac{1}{4\pi\eps_0}\,\evec_i\,\left(\d_i\d_j\frac{p_j(\tau)}{r}\right)_{(0)}\ .
  \eeqan
Note that  the second term proportional to $\ddot{\pvec}$ has no $\delta$-singularity. Equation \eref{3.5a} gives a result similar to that from the static case:
 \beqan
 \left(\Evec^{(1,\,p)}(\rvec,t)\right)_{(0)}=-\frac{1}{3\eps_0}\,\pvec(t)\,\delta(\rvec)\ .
\eeqan
The magnetic field $\Bvec^{(1,\,p)}(\rvec,t)$ of the electric dipole has no $\delta$-singularity.\\
Obviously, the electric field $\Evec^{(1,\,m)}(\rvec,t)$ has no $\delta$-singularity either. Writing the singular part of $\Bvec^{(1,\,m)}(\rvec,t)$,
\beqan
\left(\Bvec^{(1,\,m)}(\rvec,t)\right)_{(0)}=\frac{\mu_0}{4\pi}\left[\nablav\left(\nablav\cdot\frac{\mvec(\tau)}{r}\right)-\frac{1}{c^2}\frac{\ddot{\mvec}(\tau)}{r}+4\pi\,\mvec(t)\delta(\rvec)
\right]_{(0)}\ ,
\eeqan
one obtains the result similar to that from the static case:
 \beqan
 \left(\Bvec^{(1,\,m)}(\rvec,t)\right)_{(0)}=\frac{2\mu_0}{3}\mvec(t)\,\delta(\rvec)\ .
 \eeqan
Let us consider the electric field $\Evec^{(2,\,p)}(\rvec,t)$ and search the corresponding $\delta$-singularities by  introducing firstly the {\bf STF} moment $\pct^{(2)}$:
\beqan
\fl\left(\Evec^{(2,\,p)}(\rvec,t)\right)_{(0)}=-\frac{1}{8\pi\eps_0}\left[\nablav^3\vert\vert\frac{\psft^{(2)}(\tau)}{r}-\frac{1}{c^2}\nablav\vert\vert\frac{\ddot{\psft}^{(2)}(\tau)}{r}\right]_{(0)}\\
\fl\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=-\frac{1}{8\pi\eps_0}\left[\nablav^3\vert\vert\frac{\pct^{(2)}(\tau)}{r}-\frac{1}{c^2}\nablav\vert\vert\frac{\ddot{\pct}^{(2)}(\tau)}{r}+\nablav\left(\Delta\frac{\lasf(\tau)}{r}\right)-\frac{1}{c^2}\nablav\frac{\ddot{\lasf}(\tau)}{r}\right]_{(0)}\ .
\eeqan
The insertion of equation \eref{5.2} gives
\beqan
\Evec^{(2,\,p)}_{(0)}=-\frac{1}{8\pi\eps_0}\left[\nablav^3\vert\vert\frac{\pct^{(2)}(\tau)}{r}-\frac{1}{c^2}\nablav\vert\vert\frac{\ddot{\pct}^{(2)}(\tau)}{r}-4\pi\,\lasf(t)\nablav\delta(\rvec)\right]_{(0)}\ .
\eeqan
Expressing the singularity of the first term from the parenthesis by the insertion of equation \eref{3.5b}, we can finally write
  \beqa\label{5.4}
\Evec^{(2,\,p)}(\rvec,t)=\left(\Evec^{(2,\,p)}(\rvec,t)\right)_{r\ne0}+\frac{1}{5\eps_0}\pct^{(2)}(t)\vert\vert\nablav\delta(\rvec)+\frac{1}{2\eps_0}\lasf(t)\delta(\rvec)\ ,
\eeqa
where
  \beqan
\left(\Evec^{(2,\,p)}(\rvec,t)\right)_{r\ne0}=-\frac{1}{8\pi\eps_0}\left[\nablav^3\vert\vert\frac{\pct^{(2)}(\tau)}{r}-\frac{1}{c^2}\nablav\vert\vert\frac{\ddot{\pct}^{(2)}(\tau)}{r}\right]_{r\ne0}\ .
\eeqan
The result \eref{5.4} is similar to the result \eref{3.10} from the static case. For the electric 4-polar term, the electric field expression for $r\ne 0$ is invariant to the substitution $\psft^{(2)}\,\to\,\pct^{(2)}$ as in the static case but, as we will see in the following, such property  is not yet  verified for higher order multipoles in the dynamical case.\\
Searching the singularities corresponding to the electric field $\Evec^{(2,\,m)}$ of the magnetic quadrupole, the difference from the static case is obvious. Let us express this field with the help of  the {\bf STF} magnetic moment $\mct^{(2)}=\mlrt^{(2)}$:
\beqan
\fl\Evec^{(2,\,m)}(\rvec,t)=\frac{1}{8\pi\eps_0c^2}\,\nablav\times\left(\nablav\vert\vert\frac{\dot{\msft}^{(2)}}{r}\right)=\frac{1}{8\pi\eps_0c^2}\left[\nablav\times\left(\nablav\vert\vert\frac{\dot{\mct}^{(2)}}{r}\right)\right.\\
\fl+\left.\frac{1}{2}\evec_i\eps_{ijk}\eps_{lkq}\,\d_j\d_l\frac{\dot{\Nsf}_q(\tau)}{r}
\right]=\frac{1}{8\pi\eps_0c^2}\left[\nablav\times\left(\nablav\vert\vert\frac{\dot{\mct}^{(2)}}{r}\right)
-\frac{1}{2}\nablav^2\vert\vert\frac{\dot{\bbox{N}}(\tau)}{r}+\frac{1}{2}\Delta\frac{\dot{\bbox{N}}(\tau)}{r}
\right]\ .
\eeqan
We employed the same notation as in equation \eref{4.6}. The singularities added by the extension of this expression to the entire space are the following:
 \beqa\label{5.6}
\fl\;\;\;\;\;\;\;\left[\nablav\times\left(\nablav\vert\vert\frac{\dot{\mct}^{(2)}}{r}\right)\right]_{(0)}=
\evec_i\eps_{ijk}\left[\d_j\d_l\frac{\dot{\mc}_{lk}}{r}\right]_{(0)}=-\frac{4\pi}{3}\evec_i\eps_{ijk}\dot{\mc}_{jk}(t)\delta(\rvec)=0\ ,\nonumber\\
\fl\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[\nablav^2\vert\vert\frac{\dot{\bbox{N}}(\tau)}{r}\right]_{(0)}=-\frac{4\pi}{3}\dot{\bbox{N}}(t)\,\delta(\rvec)\ .
\eeqa
For  the last term from the expression of $\Evec^{(2,\,m)}$, we have to consider the equation \eref{5.2}, i.e.\
\beqan
\Delta\frac{\dot{\bbox{N}}(\tau)}{r}=\frac{1}{c^2}\frac{\tdot{\bbox{N}}(\tau)}{r}-4\pi\dot{\bbox{N}}(t)\,\delta(\rvec)\ ,
\eeqan
such that we can write
\beqa\label{5.7}
\fl\Evec^{(2,\,m)}(\rvec,t)=\frac{1}{4\pi\eps_0} \left[\frac{1}{2c^2}\nablav\times\left(\nablav\vert\vert\frac{\dot{\mct^{(2)}}(\tau)}{r}\right) - \left(\frac{1}{4c^2}\nablav^2\vert\vert\frac{\dot{\bbox{N}}(\tau)}{r}-\frac{1}{4c^4}\frac{\tdot{\bbox{N}}(\tau)}{r}\right)\right]_{r\ne0} \nonumber \\
+ \frac{1}{12\eps_0c^2}\dot{\bbox{N}}(t)\delta(\rvec)-\frac{1}{4\eps_0 c^2}\dot{\bbox{N}}(t)\delta(\rvec)\ .
\eeqa
In the above  result one can notice that for $r\ne0$, the expression of the electric field is not invariant to the substitution $\psft^{(2)}\,\to\,\pct^{(2)}$. The additional  term which appears as a consequence of this substitution is exactly the expression of the electric field corresponding to an electric dipole with the moment
 \beqa\label{5.8}
\delta\pvec^{\,'}=-\frac{1}{4c^2}\dot{\bbox{N}}\ .
\eeqa
In equation \eref{5.7}, we have written separately the term  which represents the $\delta$-singularity corresponding to this dipole. \\
Let us express the field $\Bvec^{(2)}(\rvec,t)$. For the part $\Bvec^{(2,\,p)}$, we insert the {\bf STF} moment $\pct^{(2)}$ which gives
 \beqa\label{5.9b}
\fl\Bvec^{(2,\,p)}(\rvec,t)&=&\frac{\mu_0}{8\pi}\nablav\times\left(\nablav\vert\vert\frac{\dot{\psft}^{(2)}(\tau)}{r}\right)
=\frac{\mu_0}{8\pi}\left[\nablav\times\left(\nablav\vert\vert\frac{\dot{\pct}^{(2)}(\tau)}{r}\right)+\evec_i\eps_{ijk}\d_j\d_l\frac{\lasf(\tau)}{r}\delta_{lk}\right]\nonumber\\
\fl&=&\frac{\mu_0}{8\pi}\nablav\times\left(\nablav\vert\vert\frac{\dot{\pct}^{(2)}(\tau)}{r}\right)\ ,
\eeqa
since $\eps_{ijk}\d_j\d_l\delta_{lk}=\eps_{ijk}\d_j\d_k=0$. It follows that $\Bvec^{(2,\,p)}$ is invariant to the substitution $\psft^{(2)}\,\to\,\pct^{(2)}$ and has no $\delta$-singularities due to equation \eref{5.6} written for  $\mct^{(2)}\,\to\,\pct^{(2)}$.\\
The introduction of the {\bf STF} magnetic moment $\mct^{(2)}=\mlrt^{(2)}$ in the expression of $\Bvec^{(2,\,m)}$ results in
\beqan
\fl\Bvec^{(2,\,m)}(\rvec,t)&=&-\frac{\mu_0}{8\pi}\left(\nablav^3\vert\vert\frac{\msft^{(2)}(\tau)}{r}-\nablav\vert\vert\Delta\frac{\msft^{(2)}(\tau)}{r}\right)\\
\fl&=&-\frac{\mu_0}{8\pi}\left(\nablav^3\vert\vert\frac{\mct^{(2)}(\tau)}{r}
-\nablav\vert\vert\Delta\frac{\mct^{(2)}(\tau)}{r}+\frac{1}{2}\nablav\times\Delta\frac{\bbox{N}(\tau)}{r}\right)\ .
\eeqan
Separating the $\delta$-singularities with the help of  equations \eref{3.5b} and \eref{5.2}, we obtain
 \beqa\label{5.9}
\fl\Bvec^{(2,\,m)}(\rvec,t)&=&-\frac{\mu_0}{8\pi}\left(\nablav^3\vert\vert\frac{\mct^{(2)}(\tau)}{r}-\frac{1}{c^2}\nablav\vert\vert\frac{\ddot{\mct}^{(2)}(\tau)}{r}+\frac{1}{2c^2}\nablav\times\frac{\ddot{\bbox{N}}(\tau)}{r}\right)_{r\ne 0}\nonumber\\
\fl&&-\frac{3\mu_0}{10}\mct^{(2)}(t)\vert\vert\nablav\delta(\rvec)-\frac{\mu_0}{4}\bbox{N}(t)\times\nablav\,\delta(\rvec)\ .
\eeqa
Since for $r\ne0$ the magnetic field of the magnetic quadrupole  can be written as
\beqan
\left(\Bvec^{(2,\,m)}(\rvec,t)\right)_{r\ne0}=-\frac{\mu_0}{8\pi}\left(\nablav^3\vert\vert\frac{\msft^{(2)}(\tau)}{r}-\frac{1}{c^2}\nablav\vert\vert\frac{\ddot{\msft}^{(2)}(\tau)}{r}\right)_{r\ne0}\ ,
\eeqan
we can conclude that for $r\ne0$ this field is not invariant to the substitution $\msft^{(2)}\,\to\,\mct^{(2)}$ and the additional term introduced by this substitution is equivalent to that of an electric dipole having the moment $\delta\pvec^{\,'}$ given by equation \eref{5.8}. Obviously, the corresponding term has no $\delta$-singularity such that it is not represented by a singular term in equation \eref{5.9}.\\
Let us go further to the field of  the third order electric multipole. Beginning with the electric field, we have to extend to the entire space the expression
 \beqa\label{5.9a}
\Evec^{(3,\,p)}(\rvec,t)=\frac{1}{24\pi\eps_0}\left(\nablav^4\vert\vert\frac{\psft^{(3)}(\tau)}{r}
-\frac{1}{c^2}\nablav^2\vert\vert\frac{\ddot{\psft}^{(3)}(\tau)}{r}\right)\ .
\eeqa
We consider separately each term from the last equation introducing the {\bf STF} moment $\pct^{(3)}$:
\beqan
\fl\nablav^4\vert\vert\frac{\psft^{(3)}(\tau)}{r}=\nablav^4\vert\vert\frac{\pct^{(3)}(\tau)}{r}
+\evec_i\d_i\,\d_{i_1}\d_{i_2}\d_{i_3}\frac{\delta_{\{i_1i_2}\lasf_{k\}}(\tau)}{r}
=\nablav^4\vert\vert\frac{\pct^{(3)}(\tau)}{r}+3\evec_i\d_i\d_j\frac{\lasf_j(\tau)}{r}\ .
\eeqan
Inserting equation \eref{D9} for the $\delta$-singularity of the first term from the right-hand side of the last equation and employing equation \eref{5.2} for the second one, we can write
 \beqa\label{5.10}
\fl\;\;\;\;\;\;\;\;\;\nablav^4\vert\vert\frac{\psft^{(3)}(\tau)}{r}&=&\left(\nablav^4\vert\vert\frac{\pct^{(3)}(\tau)}{r}
+\frac{3}{c^2}\nablav^2\vert\vert\frac{\ddot{\bbox{\Lambda}}(\tau)}{r} \right)_{r\ne 0}\nonumber\\
\fl\;\;\;\;\;\;\;\;\;&&-\frac{12\pi}{7}\pct^{(3)}(t)\vert\vert\nablav^2\delta(\rvec)-\frac{4\pi}{c^2}\ddot{\bbox{\Lambda}}(t)\delta(\rvec)
-12\pi\bbox{\Lambda}(t)\vert\vert\nablav^2\delta(\rvec)\ ,
\eeqa
where $\bbox{\Lambda}=\lasf_i\evec_i$. Regarding the second term from equation \eref{5.9a}, we similarly obtain
 \beqa\label{5.11}
\fl\;\;\;\;\;\;\; \nablav^2\vert\vert\frac{\ddot{\psft}^{(3)}(\tau)}{r}=\left(\nablav^2\vert\vert\frac{\ddot{\pct}^{(3)}(\tau)}{r}
 +2\,\nablav^2\vert\vert\frac{\ddot{\bbox{\Lambda}}(\tau)}{r}+\frac{1}{c^2}\frac{\qdot{\bbox{\Lambda}}(\tau)}{r}
 \right)_{r\ne 0}-\frac{20\pi}{3}\ddot{\bbox{\Lambda}}(t)\delta(\rvec)
 \eeqa
since the first term containing $\nablav^2\vert\vert(\pct^{(3)}(\tau)/r)$ has no $\delta$-singularity.
The insertion of equations \eref{5.10} and \eref{5.11} in equation \eref{5.9a} gives
\beqan
\fl\Evec^{(3,\,p)}(\rvec,t)&=&\frac{1}{24\pi\eps_0}\left(\nablav^4\vert\vert\frac{\pct^{(3)}(\tau)}{r}
-\frac{1}{c^2}\nablav^2\vert\vert\frac{\ddot{\pct}^{(3)}(\tau)}{r}\right)_{r\ne 0}\nonumber\\
&&+\frac{1}{4\pi\eps_0}\left(\frac{1}{6c^2}\nablav^2\vert\vert\frac{\ddot{\bbox{\Lambda}}(\tau)}{r}-\frac{1}{6c^4}\frac{\qdot{\bbox{\Lambda}}(\tau)}{r}\right)_{r\ne 0}\nonumber\\
&&
-\frac{1}{14\eps_0}\pct^{(3)}(t)\vert\vert\nablav^2\delta(\rvec)
-\frac{1}{2\eps_0}\bbox{\Lambda}(t)\vert\vert\nablav^2\delta(\rvec)+\frac{1}{9\eps_0}\ddot{\Lambda}(t)\delta(\rvec)\ .
\eeqan
From the above equation  we can conclude that $\Evec^{(3,\,p)}(\rvec,t)$ is not invariant to the substitution $\pct^{(3)}\,\to\,\pct^{(3)}$. This substitution introduces an additional term which has precisely the expression of the electric field of  an electric dipole of moment
 \beqa\label{5.13}
 \bbox{\delta\pvec}^{''}=\frac{1}{6c^2}\ddot{\Lambda}(t)\ .
 \eeqa
Let us express the magnetic field $\Bvec^{(3,\,p)}$:
 \beqa\label{5.14}
\Bvec^{(3,\,p)}(\rvec,t)=\frac{\mu_0}{24\pi}\nablav\times\left(\nablav^2\vert\vert\frac{\dot{\psft}^{(3)}(\tau)}{r}\right)=\frac{\mu_0}{24\pi}\evec_i\eps_{ijk}\d_j\d_{i_1}\d_{i_2}\frac{\psf_{i_1i_2k}(\tau)}{r}\ .
 \eeqa
The introduction of the {\bf STF} moment $\pct^{(3)}$ has as result
\beqan
\Bvec^{(3,\,p)}(\rvec,t)=\frac{\mu_0}{24\pi}\nablav\times\left(\nablav^2\vert\vert\frac{\dot{\pct}^{(3)}(\tau)}{r}\right)+\frac{\mu_0}{24\pi}\evec_i\eps_{ijk}\d_j\d_{i_1}\d_{i_2}\frac{\delta_{\{i_1i_2}\lasf_{k\}}(\tau)}{r}
\eeqan
and since $\nablav\times(\nablav^2\vert\vert(\dot{\pct}^{(3)}(\tau)/r))$ has no $\delta$-singularities, we can write
after simple algebraic calculation and after using equation \eref{5.2} that
 \beqan\label{5.14a}
\Bvec^{(3,\,p)}(\rvec,t)&=&\left(\frac{\mu_0}{24\pi}\nablav\times\big(\nablav^2\vert\vert\frac{\dot{\pct}^{(3)}(\tau)}{r}+\frac{\mu_0}{4\pi}\;\frac{1}{6c^2}\nablav\times\frac{\tdot{\bbox{\Lambda}}(\tau)}{r}\right)_{r\ne 0}\nonumber\\ &&+\frac{\mu_0}{6c^2}\dot{\bbox{\Lambda}}(t)\times\nablav\delta(\rvec) \ .
 \eeqan
For $r\ne 0$, the additional term introduced by the substitution $\psft^{(3)}\to\pct^{(3)}$ in equation \eref{5.14} represents the magnetic field corresponding to the electric dipole with the moment $\bbox{\delta\pvec}^{''}$ defined by equation \eref{5.13}.\\
Let us consider  the electric moment up to $n=3$, and the magnetic one up to $n=2$ and the corresponding sums of fields   representing one of the first  approximations of a complex system assimilated  with a point-like multipolar system. From the above results, as it was expected, we see that the substitutions of the primitive moments by the corresponding {\bf STF} ones introduce  some additional terms. The sum of these terms is equivalent to the substitution
\beqan
\pvec\,\to\, \widetilde{\pvec}=\pvec+\delta\pvec\ ,
\eeqan
where
\beqan
\delta\pvec=\bbox{\delta\pvec}'+\bbox{\delta\pvec}''=-\frac{1}{c^2}\dot{\bbox{t}},
\eeqan
and the vector $\bbox{t}$ is defined as
\beqan
\bbox{t}=\frac{1}{4}\Nsft-\frac{1}{6}\dot{\latens}=\frac{1}{10}\int_{\dom}\rmd^3x\, \left((\rvec.\jvec)\,\rvec-2r^2\,\jvec\right)\ .
\eeqan
This vector is the so-called electric dipolar toroidal moment and represents a first term from a series of toroidal moments introduced by Dubovik {\it et al} \cite{Dubovik-FEC}-\cite{RV} by  generalizing    Zeldovich's original idea that a closed toroidal current represents a certain new kind of dipole   \cite{Zeldovich}. The corresponding $\delta$-singularities can be easily identified in the expressions  established above.\\
The procedure can be continued for the next orders, the technique being the same as in the case of the first orders treated above. Unlike in the static case, in the dynamic one it is more complicated to establish expressions of the singularities for an arbitrary $n$. The recursive character involving a general number of $n$ steps makes such relations hard to derive and apply \cite{cv02}. Furthermore, at the current level of applications, we believe that it is mandatory to have expressions up to $n=3$.

\section{Conclusion}\label{conclusion}
In this paper we have expressed the $\delta$-singularities of the electric and magnetic multipoles. The results were   given for arbitrary multipole orders $n$ in the static case, while in the dynamic one, the outcomes correspond to the lower orders.  However, the calculation algorithm is presented in a manner that   facilitates the processing of the results step by step to the next orders of the dynamic expressions. The central idea of  the article is that, instead of employing the $\delta$-singularities of the functions $f(\tau)/r$, it is more efficient for the higher orders  to search directly such singularities for the multipole fields represented in terms of the {\bf STF} moments. Working in Cartesian coordinates and employing a particular system of parameters and notations adapted to the technique of reducing the primitive Cartesian moments to the corresponding {\bf STF} ones, we have stressed the significance of the first type of moments in the process of searching the field singularities.\\
In our opinion, a first field of application of these results is to extend the classical argumentation of the generalization of the Fermi contact term to arbitrary multipole-multipole interactions, useful in the studies of the hyperfine interaction, \cite{Bucher,Karl}.

redpajama_set_name: RedPajamaArXiv


text:
Q: How to create a CSR with a UPN SAN using python cryptography module I am trying to generate a CSR with a UPN SAN using the python cryptography module
from cryptography import x509
from cryptography.hazmat.primitives import hashes, serialization
from cryptography.hazmat.primitives.asymmetric import rsa
from cryptography.hazmat._oid import ObjectIdentifier

key = rsa.generate_private_key(public_exponent=65537, key_size=3072)

with open("rsa.key", "wb") as f:
    f.write(key.private_bytes(
        encoding=serialization.Encoding.PEM,
        format=serialization.PrivateFormat.TraditionalOpenSSL,
        encryption_algorithm=serialization.NoEncryption(),
    ))

csr = x509.CertificateSigningRequestBuilder()

csr = csr.add_extension(
    x509.SubjectAlternativeName([
        x509.OtherName(ObjectIdentifier('1.3.6.1.4.1.311.20.2.3'), "upn@email.com".encode("utf-8")),
    ]),
    critical=False
)

csr = csr.sign(key, hashes.SHA256())

But I get this error :
  csr = csr.sign(key, hashes.SHA256())
  File "/usr/local/lib/python3.8/dist-packages/cryptography/x509/base.py", line 641, in sign
    return rust_x509.create_x509_csr(self, private_key, algorithm)
ValueError: OtherName value must be valid DER: ParseError { kind: ShortData }

How can convert a string to DER binary format ?

A: ASN.1 strings have types, but assuming UTF8String is acceptable then the valid DER encoding for your example email would be: \x0c\rupn@email.com. You can use an ASN.1 DER encoder to do this, but as long as you keep the string length under 128 then the first byte is the ASN.1 tag itself and the second byte is the length (length encoding becomes multi-byte if you exceed 127 bytes). So, assuming your values are < 128 you can take that example and modify the byte value of the length of the string to obtain the value you need (e.g. upn@email.com is 13 bytes long so the DER string encodes \r, which has ordinal value 13).

A: I found the python asn1 module to do the DER encoding
import asn1

encoder = asn1.Encoder()
encoder.start()
encoder.write('upn@email.com', asn1.Numbers.UTF8String)
print(encoder.output())

redpajama_set_name: RedPajamaStackExchange


Just-in-time reporting from the UNCTAD e-commerce week!
Today marks the start of the UNCTAD E-Commerce Week 2018, a week-long event that brings together stakeholders to discuss development opportunities and challenges associated with the evolving digital economy. DiploFoundation and the Geneva Internet Platform will be actively engaged in the event.
The GIP Digital Watch observatory is providing just-in-time reporting from most sessions at the UNCTAD E-commerce Week 2018 and from the second meeting of the Intergovernmental Group of Experts on E-commerce and the Digital Economy (18–20 April), with the aim of enabling the community to keep track of discussions happening simultaneously.
Session reports are available by the end of each day. Read the reports at dig.watch/unctad2018 or follow us on @genevagip for links to reports.
On Wednesday, 18th April, we will be hosting a session on Introduction to the course on digital commerce and emerging technologies, between 08:30 and 10:00, Room XXV (Palais des Nations). The session is based on the curriculum of the online course on digital commerce, jointly delivered by DiploFoundation, CUTS International Geneva, the International Trade Centre (ITC), UNCTAD and the Geneva Internet Platform (GIP) in 2017 and 2018. The publication 'Digital commerce capacity development: preparing trade professionals for the challenges of the digital economy' will be officially launched during this session.
Some digital commerce course participants shared their feedback about the capacity building experience. Watch them online.
The GIP is an initiative supported by the Swiss authorities and operated by DiploFoundation.


text:
Help: Trouble w chords in RISE UP SINGING??
NOT IN THE BOOK
Rise Up Mudcat Songbook - Australia (860)
Rise Up Mudcat!, RUS Volume 3 (online) (58)
Rise Up Mudcat! - Preschool Songs (7)
Rise Up Singing Book II: 'Rise Again' (161)
homage to Rise Up Singing (374)
Mudcat Up Singing - a perfect songbook (34)
Lyr Req: Not in the Book (20)
blue books revisited (Rise Up Singing) (63)
Cheapest copies of RUS? (7)
RISE UP SINGING II - Current Status??? (14) (closed)
Revised RUS due next fall (9)
Help: Rise Up Singing II (10) (closed)
9/11 NYC Help--Rise Up Singing (8)
What's RISE UP SINGING? (42)
Help: Rise Up Singing Two (9) (closed)
Help: Update on 'Rise up Singing' 2000 (23)
Any news on the Rise Up Singing sequel? (18)
Sequel to Rise Up Singing coming in Spring (2) (closed)
Rise Up Singing (47)
Suggestions for Rise Up Singing II (39) (closed)
In defense of RUS (4)
WyoWoman 08 Sep 00 - 07:30 PM
catspaw49 08 Sep 00 - 07:34 PM
Midchuck 08 Sep 00 - 07:35 PM
GUEST,Barry Finn 08 Sep 00 - 07:39 PM
dwditty 08 Sep 00 - 08:49 PM
Marion 08 Sep 00 - 09:30 PM
Naemanson 08 Sep 00 - 09:41 PM
Joe Offer 08 Sep 00 - 09:49 PM
rabbitrunning 09 Sep 00 - 01:24 AM
Sandy Paton 09 Sep 00 - 04:31 AM
Big Mick 09 Sep 00 - 09:23 AM
catspaw49 09 Sep 00 - 10:46 AM
canoer 09 Sep 00 - 12:15 PM
Catrin 09 Sep 00 - 12:24 PM
Roger in Baltimore 09 Sep 00 - 12:35 PM
Sandy Paton 09 Sep 00 - 03:44 PM
Mooh 09 Sep 00 - 04:30 PM
campfire 09 Sep 00 - 04:51 PM
Mbo 09 Sep 00 - 08:00 PM
Giac 09 Sep 00 - 08:01 PM
rabbitrunning 09 Sep 00 - 10:23 PM
Bill D 09 Sep 00 - 11:10 PM
IvanB 09 Sep 00 - 11:38 PM
WyoWoman 10 Sep 00 - 01:08 AM
Subject: Trouble w chords in RISE UP SINGING??
From: WyoWoman
Date: 08 Sep 00 - 07:30 PM
I find the 'Rise Up Singing' 'hymnal' to be useful for finding words to a bazillion songs (although I agree with someone who called it 'Rise Up Squinting', which might be more of a commentary on me than it), but for the life of me, I can't figure out what the chord notations mean. I either am missing something entirely (a grave possibility, given my level of expertise), or the notations are often just not quite right.
Anyone else have any thoughts about this?
Thenkyew,
Subject: RE: Help: Trouble w/ chords in RISE UP SINGING??
From: catspaw49
A lot of them are flat out screwed. Wrong chords, wrong time, or not enough.........I know Joebro, they do a helluva' job and its only a small op, etc, etc, etc.........Work the chords out for yourself Pansy or just ask.
Spaw
From: Midchuck
I vote for the chords being not quite right. It's great as a lyric source, but I'd like a good supply of whatever the guy was on who put in the chord sequences. It would keep me from getting cabin fever in the Vermont winter, that's for sure.
From: GUEST,Barry Finn
I got a pet peeve with the hymnal or the Bible as some call it, for all it's worth it best use is as kindling on a cold winter's night even a hot summer's night for that matter. OK let 'em loose Barry who'sreadyfortheflyingfuries
Wow, that's fast
Yeah Barry, I'm with ya' and I love your song about it, but some folks like it and I've given up....as long as its a "suggestion" and not a bible it Ok except for the PC lyrics and bad chords.
It's a handy reference for remembering the words to lots of songs that I probably wouldn't be carrying the lyrics for, and I appreciate the work that went into it. I have been at song circles in which no one ever took their noses out of the Book, and that's a drag (except when *I* need to use it as a cheat sheet). But the chords can be really frustrating ...
At least it's nice to know that, even though my playing still sucks, it doesn't suck as bad as i imagined ...
Well ... maybe ...
From: dwditty
WW,
A couple of years ago I started a thread about RUS which got Barry going pretty good and may even have been the inspiration for his song (unless he had it tucked away just waiting for the "perfect time" to use it).*G* I have come to use it pretty much as a reference to find words (as you say) or to learn the basic chords (although in many cases they are not even close). In that thread, Art Thieme suggested the Folksinger's Wordbook. He is absolutely right. Better songs, no attempts to change the words to make them fit the particular "whatever" of the author, and chords that not only actually work, but also sit over the words (which are in normal type). Compiled and edited Fred & Irwin Silber. ISBN 0-8256-0140-1. Probably available at Amazon, B&N, or wherever. Be sure to access their sight from Mudcat.
From: Marion
I find the RUS chords to be offbase sometimes, but sometimes they're fine, and when they're offbase at least they're a starting point for me - I find it difficult to ear out chords for a song from scratch.
And I also really like having the chords written separately from the lyrics, and the system they use. Once you figure out how long each chord written alone is supposed to last for the particular song, then you just have to follow the pattern of playing the chords in units or double units or half units (units meaning strum-down-strum-up or a measure or two measures or whatever for that particular song). I use the RUS system in my own notes about other songs.
When the chords are written above the lyrics, that makes the first verse easier but all the other verses much more difficult to follow (unless it's written out again for every verse). If you're not very familiar with a song, trying to anticipate whether the word "said" or "look" in the second verse corresponds to the same beat as the word "wrapped" in the first verse is a real pain. Also, sometimes the chord changes happen in a pause between lines, or in the middle of a drawn out syllable, or a beat before a word, which is harder to clarify with the "write above the words" system. Much easier to say for each verse: G Bm C G....
From: Naemanson
But have you played the RUS game?
At a party or gathering choose up teams with multiple people on each team. The first team picks a page and challenges the other team to sing something from that page. The teams take turns.
Scoring is optional (The idea is to get people singing, of course.)
Able to sing just the chorus: 1 Point
Able to sing the whole song: 2 Points
Add one point for each team member who sings along
Add ten points if you can get the other team to sing too.
First team to get to a bazillion points wins.
Have fun and sing.
Hi dw, I had written that song awhile before after attending a few different singing sessions where that's all that was sung & it was all sung that way to the letter & told that that's not the way it's written in RUS so it's not way it should be done. I'd had also heard from an old friend that what was once was a great singing session in San Francisco degenerated into a RUS session. I don't really sing the sung much it gets some people using me as cannon fodder & I do realize that I have a devilish & odd sense of humor at times. Barry whostillfeelsthesameaboutRiseUpScreaming
From: Joe Offer
Wyo, there are instructions for the book's system of chord notation in the front pages of Rise Up Singing. Yes, I will admit the chords are off on many songs, but at least they're an attempt. I would also like to add that in the newer editions, most of the "politically correct" alterations of lyrics have been removed. The older, wirebound editions have some pretty awful changes made to some songs.
Yes, I prefer Folksinger's Word Book. It costs $34.95, rarely discounted. Our Song Circle buys Rise Up Singing by the case from Sing Out! and we're able to resell them for under $15 and make a profit.
-Joe Offer-
From: rabbitrunning
Date: 09 Sep 00 - 01:24 AM
My edition is 1988. It's wirebound, so I guess it's time to go looking for a newer copy, huh?
I like it, to tell the truth, even tho I didn't recognize all the lyrics in there. Which ones were PC'd, and do we know by who?
I keep finding lyrics I don't recognize here, too. I guess I don't mind because I expect anything which qualifies as "folklore" to have lots of variants. The fascinating bit to me is trying to track the change over time and places. That's why I love it when people cite their sources!
From: Sandy Paton
Aside from the "screwed up" chords, the texts of the traditional songs the book offers are rarely the ones I would have selected from the many versions available in various published sources. Maybe the better texts just weren't available, but to have this mess become a song circle's Bible is a major tragedy. For Heaven's sake, Joe, don't push it just because it's cheap and you can make money selling it! It is NOT a very good source for song texts, and yet the versions it contains, because of the thousands of copies now in the hands of the innocents, are getting to be looked upon as the definitive ones.
While I'm at it, I noticed some errors in the headnotes to the songs, as well, before I put the book aside as more of an irritant than an aid. Beware of the misinformation it contains.
Sandy (a tired and grouchy old curmudgeon)
From: Big Mick
I am absolutely with the grouchy old curmudgeon and Bro Barry. There are songs that I have learned from the singing of cronies of my Grandparents that bear no resemblance to the RUS version. As a publication I put in the class of a source if nothing else works. And the "PC" versions are not even that good. I much prefer "The Folksingers Wordbook" as a starting point to tracking down a version of a song.
The issue of the RUS "style police" is a separate issue entirely. If you are going to have a text that all songs sung must follow, at least use one that is credible. But I will not participate in a circle that insists that I sing and listen to only approved versions of anything. One of the major reasons I go to a session/circle is to hear and learn other folks versions of tunes/songs.
Big Mick
Well Joe..............
Da' Phoak Phogey has spoken!!! Well done there Oh Great Phoak Phogey!!!
Seriously Joe......I think that the 4th or 5th post I ever made here was about this and everytime RUS comes up the best that can be said is that "it gets people singing" in the hope that they go on and actually do other songs and learn more about the ones they do from RUS. But if you read the forum and the stories here, that is rarely the case as it ends up as a Folk Bible which it most assuredly is not. And Sandy, curmudgeon or not, finally has hit the nail on the head. We're simply passing along misinformation in a true distortion of the folk process.
There are other fine resources as we all know, including the DT. They all may not be as easily obtainable, but we're not doing anyone any service by pushing RUS. I suggested a long time ago that we should publish (somehow) a songbook that was laid out in that fashion, but more true to the process and more accurate in its information. I know that RUS operates on a "shoestring" with minimal help as I said above and as you have often pointed out......But it is still not a reason to promote a sanitized and inaccurate piece of work. They could do better.
Still love ya' Joe........but let's face it, RUS bites.
From: canoer
This is an interesting question. IF my goal is primarily to create a "community" atmosphere by means of folk singing, and IF the most easily used resource (RUS) for accomplishing this happens to have some poor choices of lyrics, some mistaken chords, and some things which are less than respectful of the actual tradition --- then, should I toss this resource and struggle with better, but much less "group-friendly," alternatives?
Please 'scuse the horrid sentence. -- Larry C.
From: Catrin
Has anybody thought of publishing a Mudcat Songbook - not to compete with Aine's of course, but one in the style of RUS - only better?
From: Roger in Baltimore
For those of you who think the chords are wrong, rejoice! It means you can "hear" the music and your instrument being "out of tune." With a little experimenting with other chords, you may find one on your own that is "in better tune." You are growing on your instrument. It means you probably could pick out a song on your instrument.
Roger in Baltimore
Catrin...My post just above yours made that suggestion and we have had several discussions on this before.
Well, while you guys were off ranting and raving about RUS, I was off singing from it with a group of 25 people, outside on a beautiful Sacramento autumn evening. And every time they did a song by James Taylor or the Beatles, the singing collapsed. "The Rose" was awful, and they decided to quit singing when I arrived in the middle of "Imagine." That's the thing I dislike most about Rise Up Singing - the James Taylor songs. I like the Beatles and I suppose "The Rose" ain't a bad song, but they sure don't work for community singing.
And James Taylor? No comment. I know a lot of people who would not be pleased by my opinion of James Taylor. They might do something horrible to me.
And yes, Sandy, I think the versions they choose of traditional songs are the recorded versions. If James Taylor sang it, that's the one they chose. And yes, it grates me that some of my favorite songs are way off from the way I know 'em and it's hard to drag the group into singing the song my way. I've led campfire singing since I was in grammar school, And I never used a book or a song sheet until I joined the song circle in Sacramento about 8 years ago.
Still, the book makes songs accessible to nonsingers. It's a struggle for the musicians in our group to force people not to be tied to the book, but we do a pretty good job of it. I have to admit that the two songs I chose last night were straight from the book but they were ones the group had never sung before - "Buffalo Boy" and "Turn Your Radio On."
I'll agree that there are all sorts of things about Rise Up Singing that really bug me. The only thing is, it's the only thing I know of that works as well as it does for group singing for nonsingers. If you can give me a practical, affordable alternative, I'd love to have it.
Of the 25 people who were at our song circle last night, maybe five would be comfortable at the biweekly circle I sing with in nearby Davis. The Davis sing is wonderful, but it's a stretch for me to fit into it since I don't play a guitar and have mostly camp songs and church songs memorized. They tolerate my cheat sheets, and they seem to have grown fond of me, so I'm glad I can enjoy the challenge of singing with them. It is a real treat to be there singing with the likes of Dick and Carol Holdstock.
But what about all our Sacramento people who just wouldn't fit into the more sophisticated circle? What should we do with them?
Download a good text from Digitrad, or, if you don't particularly like the Digitrad vesion, photocopy one from any of the fine songbooks you have been acquiring of late, Joe, then run off a couple dozen copies on the copy machine. I think you can find better versions among the books in your growing library. Let your song circle learn them! You'll be doing the tradition a favor.
I may have told this story before, but some years ago, Caroline and I were on the staff at Augusta, assigned to encourage group singing among the confidence-impaired. We were leading a song-swap session in the Chapel when one of the participants asked for "Careless Love." Okay, I've been singing "Careless Love" for fifty years, so I said, "Sure," and launched into it. Zingo! Out came half a dozen RUSes, noses were immediately buried in the pages, and when my text deviated from the printed version, I heard someone cry, "Hold on! You've got it wrong!" I tried to explain that folksongs are always in a state of flux (quoting Bruno Nettl), vary from one source to another, and pointed out that I liked to sing it the way I had learned it many years before RUS was born. But then I asked one of the RUSsers to lead the song in the way they chose to do it, giving her a chance to prove to herself that she could do it, which was good, and allowing me to avoid making a somewhat rude suggestion concerning a possibly appropriate use of the paper to which they were so obviously dedicated. Had I been a presidential candidate, I would have been glad that my mental microphones were turned off. Even Caroline might have concurred: "Right! Big time!" (and she's a very kind and gentle person, far from curmudgeonish).
From: Mooh
Like others out there(here?), I had RUS before I had the internet, and I live a long way from anything that resembles a selection of folksong books. In that light RUS is/was a Godsend, but I never accepted it as THE LAST WORD in folk music. That kind of RUS fundamentalism is deviant. (Imagine if it had actual printed music in it, the RUS fundamentalists would be impossible!) It nonetheless makes for an okay "idea" book, for me anyway, but it stands on the shelf with dozens of other songbooks none of which I accept as a bible.
I see the willingness of folkies to use it bible-like as pretty strange, since I also see folkies as by-and-large not so fundamentalist.
Anyway, its cost per song is pretty low, even if the chord choices are weird (even if you can accept the key).
There's a group of books, I think the Soodlum ones, which also have some odd chords given, missing verses which are well known in the tradition (and not given as alternatives), and includes the notes which seem to me to be somewhat generic interpretations. Oh well, maybe we can't muster the jam to look beyond RUS...
Not to argue, but I like the idea of a Mudcat book, though I certainly couldn't underwrite the project.
In the meantime, I go to several sources (including RUS) but especially here, for songs. I'm glad our audiences don't mind our warped readings of folk songs, but maybe they haven't heard of RUS.
My $0.02, Cdn funds. Mooh.
I shall order the Folksinger's Wordbook anon. Joe, maybe you could get the publshers of THAT to cut you a deal???
Oh, and RiB, I actually have gotten ok at figuring out songs that stick to the I, IV, V chords, even though I'm pretty much staying in the key of A. This is why God gave us capos, right? I actually can do all the keys that don't have B chords in them, although F is still a challenge.
I'm going this afternoon to see if I can find silk-wound strings, or something like that. A friend recommended those to make me stop whining about how my fingers hurt and how I keep damping the strings ... We shall see ...
From: campfire
I sure must not be going to the right places, cuz I've never met anyone who uses the RUS in the "Bible" method that keeps getting described here. I know several people who have copies, and like my copy, the margins are filled with extra verses, alternate versions, and so on.
The RUS book is a heck of a lot more portable than the DT. We don't have electricity around the campfire, much less a phone line. And its not the only book I - or any of the people I refer to above - use AS A GUIDE. Usually, when we're actually singing, we only use it to suggest a song, in case nobody comes up with a suggestion when the last song ends. Then we do it, our way (whoever is leading).
But then again, one of the people I'm talking about is my DAD; the guy who, after I showed him my first copy of RUS, found a bunch of songs "they got it wrong, in here". But we sat up until 4 am singing together, anyhow, and I wouldn't give that back for anything. (Then he went and bought his own copy, and the tapes, because there were some songs in there he hadn't heard and wanted to learn.)
And yes, for some of the songs, the chords don't work out right. But I've found that in lots of other printed sources, too, including music books that "match" an album and cost a whole lot more.
Campfire, I've seen both. Some singarounds, you just see the book come out so someone can check words s/he doesn't know, or to find a song s/he remembers part of. But at some song circles, it absolutely IS the Bible, and there's a slavish devotion to the Book and Nothing But the Book. (Which, regardless of what book it is, would be odious.)
But good on you and your father for getting into the music that seriously. Using it as one of several resources is terrific. I'm not dissin' the book completely -- just acknowledging that it has its downside. And trying to figure out how much of what I perceive as a downside is due to it and how much is due to me...
I just bought the Folksinger's Word Book through Amazon.com, through the jumping catfish, of course, and it's now $27. Still more expensive thatn RUS, but ... the blurb says it has over 1,000 folk songs. THAT should keep me busy...
Does anyone really think its the fault of the BOOK the way people use it? Or do you think if those SAME PEOPLE had little laptops with the DT at their disposal, (for the same length of time they've had their RUS'es) they wouldn't be just as rigid with THOSE versions being "set in stone"? Is it just a case of the book being too popular for its own good? Combined with the anal-retentive attitude of a great deal of the general public?
I think it also points to a lack these days of the whole oral tradition of learning songs (or whatever); everything "we" learn these days comes out of a book. Half the women I work with can't cook ANYTHING without a recipe. Everyone remembers Grandma's Apple Pie, but nobody can make it now that she's passed on.
Another problem is that people just don't sing enough anymore. In our church a few years ago, a great number of the congregation asked if we could stop doing the "harder" hymns during the services, because most of the congregation had trouble with them. They wanted hymns on the musical level of "Jesus Loves Me".
Enough ranting - apologies to all who listened.
I'm with Joe on this one. RUS isn't a bible and was never intended to be a bible but it gets people singing.
I understand those of you who say the versions of songs in RUS are not accurate. But how does that make you different from those who treat RUS as a bible? Aren't you holding up something else and giving it that same status?
Sandy, you have been singing some of these songs since they were collected and must be singing them correctly. How many of them have you heard in a different form on someone else's recording. Have you tried to take the artist to task for changing the song? If not, why not? That recording is now doing the same thing that you have accused RUS of. It is passing an incorrect version of the song to along to generations who will think that it is correct.
All we are saaaay-eeeng
Is give peace a chance ....
Woops. Did I turn EVERYBODY pinko?
From: Mbo
Help! I'm seeing pink and I don't even drink!
From: Giac
Aha! I thought my computer had freaked out. Thanks for pointing it out, WW. Wheeeeew!
RUS: Rodents of Unusual Size (The Princess Bride)
Gawddammitt PANSY!!!! Put the friggin' web design stuff away!!!! .......geeziz...............A little knowledge is a dangerous thing and you got yersef a .44 magnum there girl!!!
Spaw (let's have a Wesson party)
If y'all are having trouble with people using the RUS book as a bible, you might have them take a gander at the introduction. The edition I have gives them permission to change the words.
From: Bill D
Sandy hit it on the nose...RUS has made 'wrong' the standard in too many places and groups....so, it makes songs 'accessable'...what if they had done a cookbook..("add one pint of salt to the soup")...or (shudder)..medical texts? "de headbone connected to de hip bone"..*grin*....
bad metaphor?..ok, but you get the point. If too many people get locked into this 'single-source' songbook, we can have generations of problems!...ah, well, all you can do is make the point, you can't make 'em listen.
Subject: Lyr Add: RISE UP SCREAMING
Aw what the hell I can always take another beating for singing this:
Rise Up Screaming by Barry Finn
Tune: Jack In The Green by Martin Graebe
A pub session or a party is a very strange thing
They're all out of fashion no more do they sing
For they read from a book or copy a tape
They imitate sounds no mortal should make
There's no sound in the kitchen, no sound in the hall
There's a murderous screech that plays off the walls
Where is the music, where are the songs
In the mouths of monsters where no sound belongs
Dead pan they look as they sing in your face
They'll spit out the words and the tunes they'll disgrace
A song will be beat o'r and over to death
And in a round robin they'll resurrect it again
No more will be heard a version that's lost
Or a variant that's rare or two songs that were crossed
The borrowing or sharing of a tune or a song
Will be according to the Bible all else will be wrong
And now for the future, it's bleak for the song
No young mortal will dare to carry it on
They'll be none around who without books can sing
Or swap without tapes or rise up singing
Copyright Barry Finn 1996
From: IvanB
For many of the songs with which I'm familiar my original source was my grandmother. To my knowledge, she never owned a song book, so the lyrics just came out of her head. They were probably just as incorrect as those we're complaining about in RUS, and, especially in my younger years I'm sure she cleaned up some of the songs for us of impressionable ages. The reality of the 'folk process' is that lyrics and tunes are bastardized every day.
We keep debating RUS over and over and probably none of us are going to be moved. But I'm of the school that says we should use whatever tools we have, and if RUS gets people singing who never had before, that's good. If some then treat it as a 'bible,' shame on them. But there are fundamentalists of every ilk and God save us from them all!
Subject: RE: Help: Trouble w chords in RISE UP SINGING??
Well, I don't have a "Wesson" color on my chart, but I think I'm making progress here ...
A little knowledge is such a dangerous thing.
You misunderstand me, Naemanson. I don't believe I said the songs in RUS are "inaccurate." What I meant to say that of the ten or fifty or a hundred versions of a song available from a variety of sources, I was sorry they hadn't selected "better" versions -- more poetic, more complete, etc. I'm talking about traditional songs here, you realize. Of the recently written songs with known composers I assume they have gathered "accurate" texts from the original sources. I may have been singing a particular version of a traditional song for many years, but I'm well aware that other versions are equally "authentic" and may appeal to someone more than the one I have chosen to learn. I really enjoy hearing and comparing different versions of a song.
There's a thread running right now about "The Foggy Dew." I sing three very different versions of that song myself, and love each of them for one reason or another. I know another version (the one I first learned some fifty years ago) that I really DON'T like and never sing anymore, and I've meant to learn a fifth version for a number of years (good intention, yet to be realized) that I think I may like even more than all the others. None of these is more "accurate" than the rest.
Ian Robb once made the point (in his SING OUT! column) that if you like a song well enough to sing it, you ought to like it well enough to LEARN it, and then you could leave the book at home. Makes sense to me.
The old folk fogey, ranting on.
OK. This time I really think I've got it. If I screw it up with this attempt, I'm going to turn off my computer, call it a night and hope Joe and Max won't crumble my cookie for this ...
sorry guys ... xo/Pansy Rue WW Twidgett, III
shee-yut
Mudcat time: 21 January 12:31 PM EST

redpajama_set_name: RedPajamaCommonCrawl


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