chore: #83 fix equations

hamilflow/models/kepler_problem/dynamics.py

@@ -18,8 +18,9 @@ def tau_of_u_exact_elliptic(
 ) -> "npt.NDArray[np.float64]":
     r"""Exact solution for tau of u in the elliptic case.
 
-    $$ \tau = -\frac{\sqrt{e^2-u^2}}{(1-e^2)(1+u)} + (1-e^2)^{-frac{3}{2}} \cdot
-    \left(\frac{pi}{2} - \arctan\frac{e^2+u}{\sqrt{(1-e^2)(e^2-u^2)}}\right)\,. $$
+    $$ \tau = -\frac{\sqrt{e^2-u^2}}{(1-e^2)(1+u)} +
+    \left(\frac{\pi}{2} - \arctan\frac{e^2+u}{\sqrt{(1-e^2)(e^2-u^2)}}\right) /
+    (1-e^2)^{\frac{3}{2}}\,. $$
     """
     cosqr, eusqrt = 1 - ecc**2, np.sqrt(ecc**2 - u**2)
     trig_numer = np.pi / 2 - np.arctan((ecc**2 + u) / np.sqrt(cosqr) / eusqrt)
@@ -35,7 +36,7 @@ def tau_of_e_plus_u_elliptic(
     $$ \tau = \frac{\pi}{(1-e^2)^\frac{3}{2}}
     - \frac{1}{(1-e)^2}\left(\frac{2(e+u)}{e}\right)^\frac{1}{2}
     - \frac{1-9e}{24(1-e)^3}\left(\frac{2(e+u)}{e}\right)^\frac{3}{2}
-    + O\left((u+e)^\frac{5}{2}\right)\,. $$
+    + O\left((e+u)^\frac{5}{2}\right)\,. $$
     """
     epu = np.sqrt(2 * (ecc + u) / ecc)
     const = np.pi / (1 - ecc**2) ** 1.5
@@ -48,9 +49,9 @@ def tau_of_e_minus_u_elliptic(
 ) -> "npt.NDArray[np.float64]":
     r"""Expansion for tau of u in the ellpitic case at $u = +e-0$.
 
-    $$$ \tau = \frac{1}{(1+e)^2}\left(\frac{2(e-u)}{e}\right)^\frac{1}{2}
+    $$ \tau = \frac{1}{(1+e)^2}\left(\frac{2(e-u)}{e}\right)^\frac{1}{2}
     - \frac{1+9e}{24(1+e)^3}\left(\frac{2(e-u)}{e}\right)^\frac{3}{2}
-    + O\left((u-e)^\frac{5}{2}\right)\,.
+    + O\left((e-u)^\frac{5}{2}\right)\,. $$
     """
     emu = np.sqrt(2 * (ecc - u) / ecc)
     return emu / (1 + ecc) ** 2 - emu**3 * (1 + 9 * ecc) / 24 / (1 + ecc) ** 3