% latex carlos.tex ; dvips -o carlos.ps carlos.dvi
\documentclass[a4]{report}
\usepackage{czech}
\usepackage{graphicx}
\begin{document}
\titlepage
\section*{P��klad 1}

Definujte funkci $$Y(l,m,\vartheta,\phi) = P^m_l(\cos \vartheta) \left\{
\begin{array}{lc}
\sin |m|\phi &~m<0 \\
\cos m \phi  &~m\ge0
\end{array}
\right.$$
Nakreslete pak pomoc� funkce {\rm plots/sphereplot} takov� ty "orbitalov�" obr�zky.
\begin{figure}[h]
\begin{center}
{\includegraphics[width=3cm]{cv2obr1.eps}}
\end{center}
%\caption{\label{Y42} $l=4,~m=2$}
\end{figure}

\section*{P��klad 2}
Uva�ujte axi�ln� symetrick� �e�en� Laplaceovy rovnice $\Delta u(\rho,z) = 0$, kter� na ose $z$ nab�v� hodnot
$$
u(\rho=0,z) = {1\over\sqrt{1+z^2}}
$$
Dopo�t�te hodtoty $u$ v cel�m prostoru a pomoc� {\rm plots/contourplot}  vykreslete
ekvipotenci�ln� ��ry v rovin� $\rho-z$.

Vyu�ijte faktu, �e
$${1\over\sqrt{1+z^2}} =
\left\{
\begin{array}{lc}
v(z) &~z<1 \\
{1\over z} v({1\over z}) &~z>1
\end{array}
\right. ~,
$$
kde
$$
v(z) = 1 - {1 \over 2^1 . 1} z^2
+{1.3 \over 2^2 . 1.2} z^4
-{1.3.5 \over 2^3 . 1.2.3} z^6
+ ... +
{(-1)^k 1.3.5...(2k-1) \over 2^k . k!} z^{2k} + ...
$$
a skute�nosti, �e pro �e�en� Laplaceovy rovnice
lze ps�t multipolov� rozvoj
$$
u(\rho,z) = \sum_{l=0}^\infty q_l  P_l(\cos \theta) \times
\left\{
\begin{array}{lc}
r^{-l-1} &~r>1 \\
r^l  &~r<1
\end{array}
\right. ~,
$$
Kdyby {\rm orthopoly} chodilo pomalu, zkuste pou��t
$$
P_l(x)  = {2l-1\over l} x P_{l-1}(x) - {l-1\over l} P_{l-2}(x)
$$
%\section*{P��klad 3}



\end{document}

