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

  %display latex
  var("G, phi, gamma, delta, t, a, b, c, d, Omega, omega, T", 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]])

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

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

\[≠wcommand{\Bold}[1]{\mathbf{#1}}\frac{{≤ft(a² δ e^{≤ft(T a\right)} + δ³ e^{≤ft(T a\right)} + 2 i \, a δ ω e^{≤ft(T a\right)} - δ ω² e^{≤ft(T a\right)} + {≤ft(-i \, ω³ sin≤ft(T δ\right) + {≤ft(δ cos≤ft(T δ\right) - a sin≤ft(T δ\right)\right)} ω² + {≤ft(-2 i \, a δ cos≤ft(T δ\right) + {≤ft(-i \, a² + i \, δ²\right)} sin≤ft(T δ\right)\right)} ω - {≤ft(a² δ + δ³\right)} cos≤ft(T δ\right) - {≤ft(a³ + a δ²\right)} sin≤ft(T δ\right)\right)} cos≤ft(T ω\right) + {≤ft(ω³ sin≤ft(T δ\right) + {≤ft(i \, δ cos≤ft(T δ\right) - i \, a sin≤ft(T δ\right)\right)} ω² + {≤ft(2 \, a δ cos≤ft(T δ\right) + {≤ft(a² - δ²\right)} sin≤ft(T δ\right)\right)} ω + {≤ft(-i \, a² δ - i \, δ³\right)} cos≤ft(T δ\right) + {≤ft(-i \, a³ - i \, a δ²\right)} sin≤ft(T δ\right)\right)} sin≤ft(T ω\right)\right)} cos≤ft(ɸ\right) + {≤ft(a³ e^{≤ft(T a\right)} + a δ² e^{≤ft(T a\right)} + a ω² e^{≤ft(T a\right)} + i \, ω³ e^{≤ft(T a\right)} + {≤ft(i \, a² e^{≤ft(T a\right)} - i \, δ² e^{≤ft(T a\right)}\right)} ω + {≤ft(-i \, ω³ cos≤ft(T δ\right) - {≤ft(a cos≤ft(T δ\right) + δ sin≤ft(T δ\right)\right)} ω² + {≤ft(2 i \, a δ sin≤ft(T δ\right) + {≤ft(-i \, a² + i \, δ²\right)} cos≤ft(T δ\right)\right)} ω - {≤ft(a³ + a δ²\right)} cos≤ft(T δ\right) + {≤ft(a² δ + δ³\right)} sin≤ft(T δ\right)\right)} cos≤ft(T ω\right) + {≤ft(ω³ cos≤ft(T δ\right) + {≤ft(-i \, a cos≤ft(T δ\right) - i \, δ sin≤ft(T δ\right)\right)} ω² - {≤ft(2 \, a δ sin≤ft(T δ\right) - {≤ft(a² - δ²\right)} cos≤ft(T δ\right)\right)} ω + {≤ft(-i \, a³ - i \, a δ²\right)} cos≤ft(T δ\right) + {≤ft(i \, a² δ

• i \, δ³\right)} sin≤ft(T δ\right)\right)} sin≤ft(T

ω\right)\right)} sin≤ft(ɸ\right)}{a⁴ e^{≤ft(T a\right)}

• 2 \, a² δ² e^{≤ft(T a\right)} + δ⁴ e^{≤ft(T

a\right)} + ω⁴ e^{≤ft(T a\right)} + 2 \, {≤ft(a² e^{≤ft(T a\right)} - δ² e^{≤ft(T a\right)}\right)} ω²}\]

\[≠wcommand{\Bold}[1]{\mathbf{#1}}-\frac{{≤ft(G c δ ω² e^{≤ft(T a + T γ\right)} - 2 \, {≤ft(-i \, G a c δ e^{≤ft(T a\right)} - i \, G c δ γ e^{≤ft(T a\right)}\right)} ω e^{≤ft(T γ\right)} - {≤ft(G a² c δ e^{≤ft(T a\right)} + G c δ³ e^{≤ft(T a\right)} + 2 \, G a c δ γ e^{≤ft(T a\right)} + G c δ γ² e^{≤ft(T a\right)}\right)} e^{≤ft(T γ\right)} + {≤ft(G c γ³ sin≤ft(T δ\right) + i \, G c ω³ sin≤ft(T δ\right) + {≤ft(G c δ cos≤ft(T δ\right) + 3 \, G a c sin≤ft(T δ\right)\right)} γ² - {≤ft(G c δ cos≤ft(T δ\right) + 3 \, G a c sin≤ft(T δ\right) + 3 \, G c γ sin≤ft(T δ\right)\right)} ω² + {≤ft(2 \, G a c δ cos≤ft(T δ\right) + {≤ft(3 \, G a² c + G c δ²\right)} sin≤ft(T δ\right)\right)} γ + {≤ft(-2 i \, G a c δ cos≤ft(T δ\right) - 3 i \, G c γ² sin≤ft(T δ\right) - 2 \, {≤ft(i \, G c δ cos≤ft(T δ\right) + 3 i \, G a c sin≤ft(T δ\right)\right)} γ + {≤ft(-3 i \, G a² c - i \, G c δ²\right)} sin≤ft(T δ\right)\right)} ω + {≤ft(G a² c δ + G c δ³\right)} cos≤ft(T δ\right) + {≤ft(G a³ c + G a c δ²\right)} sin≤ft(T δ\right)\right)} e^{≤ft(i \, T ω\right)}\right)} cos≤ft(ɸ\right) + {≤ft(-i \, G c ω³ e^{≤ft(T a + T γ\right)} + 3 \, {≤ft(G a c e^{≤ft(T a\right)} + G c γ e^{≤ft(T a\right)}\right)} ω² e^{≤ft(T γ\right)} + {≤ft(3 i \, G a² c e^{≤ft(T a\right)} + i \, G c δ² e^{≤ft(T a\right)} + 6 i \, G a c γ e^{≤ft(T a\right)} + 3 i \, G c γ² e^{≤ft(T a\right)}\right)} ω e^{≤ft(T γ\right)} - {≤ft(G a³ c e^{≤ft(T a\right)} + G a c δ² e^{≤ft(T a\right)} + 3 \, G a c γ² e^{≤ft(T a\right)} + G c γ³ e^{≤ft(T a\right)}

• {≤ft(3 \, G a² c e^{≤ft(T a\right)} + G c δ² e^{≤ft(T

a\right)}\right)} γ\right)} e^{≤ft(T γ\right)} + {≤ft(G c γ³ cos≤ft(T δ\right) + i \, G c ω³ cos≤ft(T δ\right) + {≤ft(3 \, G a c cos≤ft(T δ\right) - G c δ sin≤ft(T δ\right)\right)} γ² - {≤ft(3 \, G a c cos≤ft(T δ\right) + 3 \, G c γ cos≤ft(T δ\right) - G c δ sin≤ft(T δ\right)\right)} ω² - {≤ft(2 \, G a c δ sin≤ft(T δ\right) - {≤ft(3 \, G a² c + G c δ²\right)} cos≤ft(T δ\right)\right)} γ + {≤ft(-3 i \, G c γ² cos≤ft(T δ\right) + 2 i \, G a c δ sin≤ft(T δ\right) - 2 \, {≤ft(3 i \, G a c cos≤ft(T δ\right) - i \, G c δ sin≤ft(T δ\right)\right)} γ

• {≤ft(-3 i \, G a² c - i \, G c δ²\right)} cos≤ft(T

δ\right)\right)} ω + {≤ft(G a³ c + G a c δ²\right)} cos≤ft(T δ\right) - {≤ft(G a² c δ + G c δ³\right)} sin≤ft(T δ\right)\right)} e^{≤ft(i \, T ω\right)}\right)} sin≤ft(ɸ\right)}{ω⁴ e^{≤ft(T a + T γ\right)} - 4 \, {≤ft(-i \, a e^{≤ft(T a\right)} - i \, γ e^{≤ft(T a\right)}\right)} ω³ e^{≤ft(T γ\right)} - 2 \, {≤ft(3 \, a² e^{≤ft(T a\right)} + δ² e^{≤ft(T a\right)} + 6 \, a γ e^{≤ft(T a\right)} + 3 \, γ² e^{≤ft(T a\right)}\right)} ω² e^{≤ft(T γ\right)} - 4 \, {≤ft(i \, a³ e^{≤ft(T a\right)} + i \, a δ² e^{≤ft(T a\right)} + 3 i \, a γ² e^{≤ft(T a\right)} + i \, γ³ e^{≤ft(T a\right)} + {≤ft(3 i \, a² e^{≤ft(T a\right)} + i \, δ² e^{≤ft(T a\right)}\right)} γ\right)} ω e^{≤ft(T γ\right)} + {≤ft(a⁴ e^{≤ft(T a\right)} + 2 \, a² δ² e^{≤ft(T a\right)} + δ⁴ e^{≤ft(T a\right)} + 4 \, a γ³ e^{≤ft(T a\right)}

• γ⁴ e^{≤ft(T a\right)} + 2 \, {≤ft(3 \, a² e^{≤ft(T

a\right)} + δ² e^{≤ft(T a\right)}\right)} γ² + 4 \, {≤ft(a³ e^{≤ft(T a\right)} + a δ² e^{≤ft(T a\right)}\right)} γ\right)} e^{≤ft(T γ\right)}}\]

:END:

\[\newcommand{\Bold}[1]{\mathbf{#1}}G c e^{\left(-{\left(a + \gamma\right)} t\right)} \sin\left(\delta t + \phi\right) \sin\left(\omega t + b\right)\]

  var('t,s,r,l,u')
  var('P_k,L_k,B_n,C_n,B_m,C_m,G_l,W_l,Gc_l,Wc_l', domain=CC)

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(P_{k}, L_{k}, B_{n}, C_{n}, B_{m}, C_{m}, G_{l}, W_{l}, \mathit{Gc}_{l}, \mathit{Wc}_{l}\right)\]

  α(t) = G_l * exp(-W_l * t)
  α_conj(t) = Gc_l * exp(-Wc_l * t)
  α_dot(t) = P_k * exp(-L_k * t)
  B_1(t) = B_n * exp(-C_n * t)
  B_2(t) = B_m * exp(-C_m * t)
  α_conj

\[\newcommand{\Bold}[1]{\mathbf{#1}}t \ {\mapsto}\ \mathit{Gc}_{l} e^{\left(-\mathit{Wc}_{l} t\right)}\]

  inner = integrate(B_1(t-r-u) * α(u), u, 0, t-r) + integrate(B_1(t-r+u) * α_conj(u), u, 0, r)
  inner

\[\newcommand{\Bold}[1]{\mathbf{#1}}-B_{n} G_{l} {\left(\frac{e^{\left(C_{n} r - C_{n} t\right)}}{C_{n} - W_{l}} - \frac{e^{\left(W_{l} r - W_{l} t\right)}}{C_{n} - W_{l}}\right)} + B_{n} \mathit{Gc}_{l} {\left(\frac{e^{\left(C_{n} r - C_{n} t\right)}}{C_{n} + \mathit{Wc}_{l}} - \frac{e^{\left(-\mathit{Wc}_{l} r - C_{n} t\right)}}{C_{n} + \mathit{Wc}_{l}}\right)}\]

  assume(C_n/L_k != -1)
  assume(W_l/L_k != -1)
  assume(-Wc_l/L_k != -1)
  assume(-Wc_l/L_k != -1)
  assume(L_k/(L_k+C_m) != -1)

  whole = (B_2(s-r) * α_dot(t-s) * inner).integrate(r, 0, s).simplify_full()

  %display plain
  integ = whole.integrate(s, 0, t, algorithm='giac')

  import sympy
  from sympy.utilities.codegen import codegen
  integ_s = sympy.sympify(integ)
  result = codegen(("conv_part", integ_s), "F95")
  for name, contents in result:
      with open(name, 'w') as f:
          f.write(contents)

  %display latex
  (B_2(s-r) * α_dot(t-s)).integrate(s, r, t)

\[\newcommand{\Bold}[1]{\mathbf{#1}}-B_{m} P_{k} {\left(\frac{e^{\left(C_{m} r - C_{m} t\right)}}{C_{m} - L_{k}} - \frac{e^{\left(L_{k} r - L_{k} t\right)}}{C_{m} - L_{k}}\right)}\]

  %display latex
  (B_1(t-r-u) * α(u)).integrate(u, 0, t-r)

\[\newcommand{\Bold}[1]{\mathbf{#1}}-B_{n} G_{l} {\left(\frac{e^{\left(C_{n} r - C_{n} t\right)}}{C_{n} - W_{l}} - \frac{e^{\left(W_{l} r - W_{l} t\right)}}{C_{n} - W_{l}}\right)}\]

  %display latex
  (B_1(t-r+u) * α_conj(u)).integrate(u, 0, r)

\[\newcommand{\Bold}[1]{\mathbf{#1}}B_{n} \mathit{Gc}_{l} {\left(\frac{e^{\left(C_{n} r - C_{n} t\right)}}{C_{n} + \mathit{Wc}_{l}} - \frac{e^{\left(-\mathit{Wc}_{l} r - C_{n} t\right)}}{C_{n} + \mathit{Wc}_{l}}\right)}\]

  assume(C_n/C_m != -1)

  assume(L_k/C_m+C_n/C_m-1 != -1)

  ((exp(-C_m * r)) * (exp(-W_l * r))).integrate(r, 0, t)

\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{e^{\left(-C_{m} t - W_{l} t\right)}}{C_{m} + W_{l}} + \frac{1}{C_{m} + W_{l}}\]

  ((exp(-L_k * (t-r))) * (exp(-Wc_l*r)*exp(-C_n*t))).integrate(r, 0, t)

  var('t, wc')
  %display latex
  diff(1/pi * (wc/(1+I*wc*t))^2, t)

\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 i \, \mathit{wc}^{3}}{\pi {\left(i \, t \mathit{wc} + 1\right)}^{3}}\]