Electricity/Electric circuit/LC circuit

/LC circuit/
Circuit at equilibrium
 * $$v_L + v_C = 0$$
 * $$L \frac{d}{dt} i + \frac{1}{C} \int i dt= 0$$
 * $$\frac{d^2}{dt^2} i = - \frac{1}{T} i$$
 * $$ i = A e^{\pm j \sqrt{\frac{1}{T}}t} = A e^{\pm j \omega t} = A Sin \omega t$$
 * $$ \omega = \sqrt{\frac{1}{T}}$$
 * $$ T = LC$$

Circuit at resonance
 * $$Z_L + Z_C = 0$$
 * $$j \omega L + \frac{1}{j \omega C} = 0$$
 * $$\omega = \pm \sqrt{\frac{1}{T}}$$
 * $$T = LC$$
 * $$V_L + V_C = 0$$
 * $$v(\theta) =A Sin(\omega + 2 \pi) - A Sin(\omega - 2 \pi)$$