#+PROPERTY: header-args :session laplace_sage :kernel sage :pandoc yes :async yes

#+begin_src jupyter-python
  %display latex
  var("G, phi, gamma, delta, t, a, b, c, d, Omega", domain=RR)
  var("z", domain=CC)
#+end_src

#+RESULTS:
:RESULTS:
\[\newcommand{\Bold}[1]{\mathbf{#1}}z\]
:END:

#+begin_src jupyter-python
  W = gamma + I*delta
  alpha(t) = G * exp(-W*t - I * phi)
  alpha
#+end_src

#+RESULTS:

#+begin_src jupyter-python
  im_alpha = (imag(alpha))
  im_alpha
#+end_src

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

#+begin_src jupyter-python
  im_alpha.laplace(t, z)
#+end_src

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


#+begin_src jupyter-python
  matrix([[z, -Omega], [Omega + a, z]]).inverse().simplify_full()
#+end_src

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


#+begin_src jupyter-python
  matrix([[0, 0], [1, 0]]) * matrix([[a, b], [c, d]])
#+end_src

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

#+begin_src jupyter-python
  matrix([[0, 1], [-1, 0]]) * matrix([[a, b], [c, d]])
#+end_src

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