master-thesis/python/energy_flow_proper/03_gaussian/laplace_sage.org

  %display latex
  var("G, phi, gamma, delta, t, a, b, c, d, Omega", domain=RR)
  var("z", domain=CC)

\[\newcommand{\Bold}[1]{\mathbf{#1}}z\]

  W = gamma + I*delta
  alpha(t) = G * exp(-W*t - I * phi)
  alpha

  im_alpha = (imag(alpha))
  im_alpha

\[\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ G e^{\left(-\gamma t\right)} \sin\left(-\delta t - \phi\right)\]

  im_alpha.laplace(t, z)

    -G⋅(δ⋅cos(φ) + γ⋅sin(φ) + z⋅sin(φ))
t ↦ ────────────────────────────────────
             2    2            2
            δ  + γ  + 2⋅γ⋅z + z

  matrix([[z, -Omega], [Omega + a, z]]).inverse().simplify_full()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \frac{z}{\Omega^{2} + \Omega a + z^{2}} & \frac{\Omega}{\Omega^{2} + \Omega a + z^{2}} \\ -\frac{\Omega + a}{\Omega^{2} + \Omega a + z^{2}} & \frac{z}{\Omega^{2} + \Omega a + z^{2}} \end{array}\right)\]

  matrix([[0, 0], [1, 0]]) * matrix([[a, b], [c, d]])

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 0 & 0 \\ a & b \end{array}\right)\]

  matrix([[0, 1], [-1, 0]]) * matrix([[a, b], [c, d]])

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} c & d \\ -a & -b \end{array}\right)\]