(I had edited this question a number of times but did not receive a satisfactory answer. So I am re-wording this question yet again. The earlier version of this question can be found here : https://docs.google.com/document/d/1_Z6iAc_LwoTxCrevoSqpoJw5TDHvVqLuDVXktPOu2_w/ )
My whole question consists of many parts (I have added a few additional doubts) :
Justifying (without skipping any logic) that current in any branch of a linear AC Circuit will be of the form $I(t)=I_0 \cos(\omega t + \phi)$. Start by writing a general set of differential equations, $n$ equations for $n$ loops and then prove that the solution will always be just a single sinusoidal term.
Now that the sinusoidal nature of currents is properly established and justified, we can represent the sinusoidal functions by complex numbers. But the actual current flowing in any branch will be the real part of the complex current. The imaginary component has no important role. It is just there. Now that the current is of the form $I_0 e^{i (\omega t + \phi)}$, we can use the formulas $V= IR$, $V= L \frac {dI}{dt}$ and $V= \frac Q C$ to find voltages across resistors, inductors and capacitors respectively. But why do we use $I=I_0 e^{i (\omega t + \phi)}$ for calculating voltages across inductors and capacitors? Shouldn't we only use $I=Re(I_0 e^{i (\omega t + \phi)})$ because only the real part of the complex current is the actual current flowing in the branch? If we use the real as well as the imaginary part of find the voltage across an inductor, then during differentiation the iota operator will "come down" and will be multiplied with the complex number and so the earlier-real part will become imaginary and the earlier-imaginary part will become real. Why is it okay to use complex current to find the voltage?
Also, while calculating power we use $S= VI^*$, where $I^*$ conjugate of complex current. Why is it not $S=VI$ or $S=Re(V)Re(I)$ ?
Any sinusoidal quantity associated with the circuit will be of the form of $Q=Q_0 e^{i(\omega t + \phi)}$. But when we apply Kirchhoff's Voltage Law we just use $Q=Q \angle {\phi}= Qe^{i \phi}$. Where did the $\omega t$ part in the complex exponential go ?
Thanks in advance :-).
EDIT 1: All of my doubts will be solved if you can point me to a reference book or something which explains everything about the basics of AC circuits in proper mathematical detail with proper justification of each and every step, calculation and assumption. Does such a book even exist ?
EDIT 2: Still no answers !