#+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, omega, T", 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:

#+begin_src jupyter-python
  assume(T>0)
  integrate(exp(I* omega * t) * exp(-t * a) * sin(delta * t + phi), t, 0, T, algorithm='giac').simplify_full()
#+end_src

#+RESULTS:
:RESULTS:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(a^{2} \delta e^{\left(T
a\right)} + \delta^{3} e^{\left(T a\right)} + 2 i \, a \delta \omega
e^{\left(T a\right)} - \delta \omega^{2} e^{\left(T a\right)} +
{\left(-i \, \omega^{3} \sin\left(T \delta\right) + {\left(\delta
\cos\left(T \delta\right) - a \sin\left(T \delta\right)\right)}
\omega^{2} + {\left(-2 i \, a \delta \cos\left(T \delta\right) +
{\left(-i \, a^{2} + i \, \delta^{2}\right)} \sin\left(T
\delta\right)\right)} \omega - {\left(a^{2} \delta + \delta^{3}\right)}
\cos\left(T \delta\right) - {\left(a^{3} + a \delta^{2}\right)}
\sin\left(T \delta\right)\right)} \cos\left(T \omega\right) +
{\left(\omega^{3} \sin\left(T \delta\right) + {\left(i \, \delta
\cos\left(T \delta\right) - i \, a \sin\left(T \delta\right)\right)}
\omega^{2} + {\left(2 \, a \delta \cos\left(T \delta\right) +
{\left(a^{2} - \delta^{2}\right)} \sin\left(T \delta\right)\right)}
\omega + {\left(-i \, a^{2} \delta - i \, \delta^{3}\right)} \cos\left(T
\delta\right) + {\left(-i \, a^{3} - i \, a \delta^{2}\right)}
\sin\left(T \delta\right)\right)} \sin\left(T \omega\right)\right)}
\cos\left(\phi\right) + {\left(a^{3} e^{\left(T a\right)} + a \delta^{2}
e^{\left(T a\right)} + a \omega^{2} e^{\left(T a\right)} + i \,
\omega^{3} e^{\left(T a\right)} + {\left(i \, a^{2} e^{\left(T
a\right)} - i \, \delta^{2} e^{\left(T a\right)}\right)} \omega +
{\left(-i \, \omega^{3} \cos\left(T \delta\right) - {\left(a \cos\left(T
\delta\right) + \delta \sin\left(T \delta\right)\right)} \omega^{2} +
{\left(2 i \, a \delta \sin\left(T \delta\right) + {\left(-i \, a^{2} +
i \, \delta^{2}\right)} \cos\left(T \delta\right)\right)} \omega -
{\left(a^{3} + a \delta^{2}\right)} \cos\left(T \delta\right) +
{\left(a^{2} \delta + \delta^{3}\right)} \sin\left(T
\delta\right)\right)} \cos\left(T \omega\right) + {\left(\omega^{3}
\cos\left(T \delta\right) + {\left(-i \, a \cos\left(T \delta\right) - i
\, \delta \sin\left(T \delta\right)\right)} \omega^{2} - {\left(2 \, a
\delta \sin\left(T \delta\right) - {\left(a^{2} - \delta^{2}\right)}
\cos\left(T \delta\right)\right)} \omega + {\left(-i \, a^{3} - i \, a
\delta^{2}\right)} \cos\left(T \delta\right) + {\left(i \, a^{2} \delta
+ i \, \delta^{3}\right)} \sin\left(T \delta\right)\right)} \sin\left(T
\omega\right)\right)} \sin\left(\phi\right)}{a^{4} e^{\left(T a\right)}
+ 2 \, a^{2} \delta^{2} e^{\left(T a\right)} + \delta^{4} e^{\left(T
a\right)} + \omega^{4} e^{\left(T a\right)} + 2 \, {\left(a^{2}
e^{\left(T a\right)} - \delta^{2} e^{\left(T a\right)}\right)}
\omega^{2}}\]
:END:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(G c \delta \omega^{2}
e^{\left(T a + T \gamma\right)} - 2 \, {\left(-i \, G a c \delta
e^{\left(T a\right)} - i \, G c \delta \gamma e^{\left(T
a\right)}\right)} \omega e^{\left(T \gamma\right)} - {\left(G a^{2} c
\delta e^{\left(T a\right)} + G c \delta^{3} e^{\left(T a\right)} + 2 \,
G a c \delta \gamma e^{\left(T a\right)} + G c \delta \gamma^{2}
e^{\left(T a\right)}\right)} e^{\left(T \gamma\right)} + {\left(G c
\gamma^{3} \sin\left(T \delta\right) + i \, G c \omega^{3} \sin\left(T
\delta\right) + {\left(G c \delta \cos\left(T \delta\right) + 3 \, G a c
\sin\left(T \delta\right)\right)} \gamma^{2} - {\left(G c \delta
\cos\left(T \delta\right) + 3 \, G a c \sin\left(T \delta\right) + 3 \,
G c \gamma \sin\left(T \delta\right)\right)} \omega^{2} + {\left(2 \, G
a c \delta \cos\left(T \delta\right) + {\left(3 \, G a^{2} c + G c
\delta^{2}\right)} \sin\left(T \delta\right)\right)} \gamma + {\left(-2
i \, G a c \delta \cos\left(T \delta\right) - 3 i \, G c \gamma^{2}
\sin\left(T \delta\right) - 2 \, {\left(i \, G c \delta \cos\left(T
\delta\right) + 3 i \, G a c \sin\left(T \delta\right)\right)} \gamma +
{\left(-3 i \, G a^{2} c - i \, G c \delta^{2}\right)} \sin\left(T
\delta\right)\right)} \omega + {\left(G a^{2} c \delta + G c
\delta^{3}\right)} \cos\left(T \delta\right) + {\left(G a^{3} c + G a c
\delta^{2}\right)} \sin\left(T \delta\right)\right)} e^{\left(i \, T
\omega\right)}\right)} \cos\left(\phi\right) + {\left(-i \, G c
\omega^{3} e^{\left(T a + T \gamma\right)} + 3 \, {\left(G a c
e^{\left(T a\right)} + G c \gamma e^{\left(T a\right)}\right)}
\omega^{2} e^{\left(T \gamma\right)} + {\left(3 i \, G a^{2} c
e^{\left(T a\right)} + i \, G c \delta^{2} e^{\left(T a\right)} + 6 i \,
G a c \gamma e^{\left(T a\right)} + 3 i \, G c \gamma^{2} e^{\left(T
a\right)}\right)} \omega e^{\left(T \gamma\right)} - {\left(G a^{3} c
e^{\left(T a\right)} + G a c \delta^{2} e^{\left(T a\right)} + 3 \, G a
c \gamma^{2} e^{\left(T a\right)} + G c \gamma^{3} e^{\left(T a\right)}
+ {\left(3 \, G a^{2} c e^{\left(T a\right)} + G c \delta^{2} e^{\left(T
a\right)}\right)} \gamma\right)} e^{\left(T \gamma\right)} + {\left(G c
\gamma^{3} \cos\left(T \delta\right) + i \, G c \omega^{3} \cos\left(T
\delta\right) + {\left(3 \, G a c \cos\left(T \delta\right) - G c \delta
\sin\left(T \delta\right)\right)} \gamma^{2} - {\left(3 \, G a c
\cos\left(T \delta\right) + 3 \, G c \gamma \cos\left(T \delta\right) -
G c \delta \sin\left(T \delta\right)\right)} \omega^{2} - {\left(2 \, G
a c \delta \sin\left(T \delta\right) - {\left(3 \, G a^{2} c + G c
\delta^{2}\right)} \cos\left(T \delta\right)\right)} \gamma + {\left(-3
i \, G c \gamma^{2} \cos\left(T \delta\right) + 2 i \, G a c \delta
\sin\left(T \delta\right) - 2 \, {\left(3 i \, G a c \cos\left(T
\delta\right) - i \, G c \delta \sin\left(T \delta\right)\right)} \gamma
+ {\left(-3 i \, G a^{2} c - i \, G c \delta^{2}\right)} \cos\left(T
\delta\right)\right)} \omega + {\left(G a^{3} c + G a c
\delta^{2}\right)} \cos\left(T \delta\right) - {\left(G a^{2} c \delta +
G c \delta^{3}\right)} \sin\left(T \delta\right)\right)} e^{\left(i \, T
\omega\right)}\right)} \sin\left(\phi\right)}{\omega^{4} e^{\left(T a +
T \gamma\right)} - 4 \, {\left(-i \, a e^{\left(T a\right)} - i \,
\gamma e^{\left(T a\right)}\right)} \omega^{3} e^{\left(T
\gamma\right)} - 2 \, {\left(3 \, a^{2} e^{\left(T a\right)} +
\delta^{2} e^{\left(T a\right)} + 6 \, a \gamma e^{\left(T a\right)} + 3
\, \gamma^{2} e^{\left(T a\right)}\right)} \omega^{2} e^{\left(T
\gamma\right)} - 4 \, {\left(i \, a^{3} e^{\left(T a\right)} + i \, a
\delta^{2} e^{\left(T a\right)} + 3 i \, a \gamma^{2} e^{\left(T
a\right)} + i \, \gamma^{3} e^{\left(T a\right)} + {\left(3 i \, a^{2}
e^{\left(T a\right)} + i \, \delta^{2} e^{\left(T a\right)}\right)}
\gamma\right)} \omega e^{\left(T \gamma\right)} + {\left(a^{4}
e^{\left(T a\right)} + 2 \, a^{2} \delta^{2} e^{\left(T a\right)} +
\delta^{4} e^{\left(T a\right)} + 4 \, a \gamma^{3} e^{\left(T a\right)}
+ \gamma^{4} e^{\left(T a\right)} + 2 \, {\left(3 \, a^{2} e^{\left(T
a\right)} + \delta^{2} e^{\left(T a\right)}\right)} \gamma^{2} + 4 \,
{\left(a^{3} e^{\left(T a\right)} + a \delta^{2} e^{\left(T
a\right)}\right)} \gamma\right)} e^{\left(T \gamma\right)}}\]
:END:
#+RESULTS:
:RESULTS:
\[\newcommand{\Bold}[1]{\mathbf{#1}}G c e^{\left(-{\left(a +
\gamma\right)} t\right)} \sin\left(\delta t + \phi\right)
\sin\left(\omega t + b\right)\]
:END: