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