Large Deformations

$$ \begin{gathered} Q_l = \frac{{4\pi }}{{2l + 3}}\int {\text{d}\zeta (} \varrho ^2 + \zeta ^2 )^{l/2 + 1} [\text{P}_l (x) - \text{P}_{l + 2} (x)] \hfill \\ \text{Where}\,x = \frac{\zeta }{{\sqrt {\varrho ^2 + \zeta ^2 } }} \hfill \\ \end{gathered} $$

$$ \begin{gathered} Q = \frac{\pi }{2}\int {\text{d}\zeta \,\varrho ^\text{2} (4\zeta ^2 - \varrho ^2 )} \hfill \\ Q_4 = \frac{\pi }{4}\int {\text{d}\zeta \text{ }\varrho ^\text{2} (8\zeta ^4 - } 12\zeta ^2 \varrho ^2 + \varrho ^4 \hfill) \\ \end{gathered} $$

$$ \begin{gathered} r^2_\text{rms = }\frac{\text{3}}{\text{8}}\int {\text{d}\zeta \varrho ^2 \text{(2}\zeta ^\text{2} + \varrho ^2 } ) \hfill \\ \overline {(\delta r)^2 } = \frac{1}{2}\int {\text{d}\zeta } \varrho \frac{{e - \zeta \varrho \prime}}{{(\varrho ^2 + \zeta ^2 )^{3/2} }}\left( {\sqrt {\varrho ^2 + \zeta ^2 - 1} } \right)^2 \hfill \\ \end{gathered} $$

$$ \begin{gathered} J_\parallel = \frac{{15}}{{16}}\int {\text{d}\zeta \varrho ^\text{4} } \hfill \\ J_ \bot = \frac{{15}}{{32}}\int {\text{d}\zeta \varrho ^\text{2} (4\zeta ^2 + \varrho ^2 )} \hfill \\ \end{gathered} $$

$$\begin{gathered} {B_{{\text{surf }}}}{\text{ = }}\frac{1}{2}\int {d\zeta } \varrho \sqrt {1 + \varrho {'^2}} {\text{ }} \hfill \\ {\text{ with }}\varrho ' = {\text{d}}\varrho (\zeta )/{\text{d}}\zeta {\text{.}} \hfill \\ \end{gathered} $$