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| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex index b540531..fb5a144 100644 --- a/buch/papers/kugel/spherical-harmonics.tex +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -871,8 +871,8 @@ computational cost lower by a factor of six \cite{davari_new_2013}. The goal of this subsection's part is to apply the recurrence relations of the $P^m_n(z)$ functions to the Spherical Harmonics. With some little adjustments we will be able to have recursion equations for them too. As previously written -the most of the work is already done. Now it is only a matter of minor -mathematical operations/rearrangements. We can start by listing all of them: +most of the work is already done. Now it is only a matter of minor mathematical +operations/rearrangements. We can start by listing all of them: \begin{subequations} \begin{align} Y^m_n(\vartheta, \varphi) &= \dfrac{1}{(2n+1)\cos \vartheta} \left[ @@ -899,7 +899,7 @@ mathematical operations/rearrangements. We can start by listing all of them: \begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-1}] We can multiply both sides of equality in \eqref{kugel:eqn:rec-leg-1} by $e^{im \varphi}$ and perform the substitution $z=\cos \vartheta$. After a few simple - algebraic steps, we will obtain the relation we are looking for + algebraic steps, we will obtain the relation we are looking for. \end{proof} \begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-2}] |