pi, tau zerolim jednoduché výrazy v Taylorově normalisaci

Former-commit-id: db59f269beb52d1c25f8a5a923cb42c14791bfb5

qpms/qpms_p.py

@@ -214,6 +214,9 @@ def zJn(n, z, J=1):
 
 
 # The following 4 funs have to be refactored, possibly merged
+
+# FIXME: this can be expressed simply as:
+# $$ -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}+\delta_{m,-1}) $$
 def π̃_zerolim(nmax): # seems OK
     """
     lim_{θ→ 0-} π̃(cos θ)
@@ -248,6 +251,8 @@ def π̃_pilim(nmax): # Taky OK, jen to možná není kompatibilní se vzorečky
     π̃_y = prenorm *     π̃_y
     return π̃_y
 
+# FIXME: this can be expressed simply as
+# $$ -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}-\delta_{m,-1}) $$
 def τ̃_zerolim(nmax):
     """
     lim_{θ→ 0-} τ̃(cos θ)

worknotes.lyx

@@ -1717,12 +1717,34 @@ Numerics:
 \end_layout
 
 \begin_layout Section
-TODO
+Misc
 \end_layout
 
 \begin_layout Itemize
-Päivi's suggestion: suppress the dipole and let it interact only with the
+The 
- higher multipoles.
+\begin_inset Quotes eld
+\end_inset
+
+zero limits
+\begin_inset Quotes erd
+\end_inset
+
+ of 
+\begin_inset Formula $\tilde{\pi},\tilde{\tau}$
+\end_inset
+
+ functions in Taylor's normalisation can be expressed as
+\lang finnish
+
+\begin_inset Formula 
+\begin{eqnarray*}
+\lim_{\theta\to0}\tilde{\pi}_{mn}\left(\cos\theta\right) & = & -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}+\delta_{m,-1})\\
+\lim_{\theta\to0}\tilde{\tau}_{mn}\left(\cos\theta\right) & = & -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}-\delta_{m,-1})
+\end{eqnarray*}
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard