I'm trying to verify the complex impedances of passive components in the phasor domain (not s-domain). Take for example the inductor, whose element law is \$ v = L \frac{di}{dt} \$. Assuming a complex exponential current drive where \$ i = I e^{j \omega t}\$ (where \$I\$ is a phasor) then we get: \$ v = L I (j \omega)e^{j \omega t} \$. From this we see that the voltage will also be a complex exponential with a phasor \$ V = (j \omega) L I \$, and so the complex impedance is \$ j \omega L \$ as expected. This works fine.
I'm trying to do the same thing assuming a complex exponential voltage instead of assuming a complex exponential current, which means I need to do an integral instead: \$ i(t) = \frac{1}{L} \int_{0}^{t} v(\tau)d \tau + i(0)\$. Plugging in \$ v(t)=Ve^{j \omega t} \$ and evaluating the integral, I get: \$ i(t) = \frac{V}{j \omega L} e^{j \omega t} + (i(0) – \frac{V}{j \omega L})\$. If it weren't for the term in parentheses then this would be the expected result. If I assume that \$ i(0) = \frac{V}{j \omega L} \$ then it works out. Is this calculation and assumption about the initial current correct, and if so is it just a mathematical requirement without a conceptual reason? Have I made an error somewhere?
Side comments: my understanding is that typically in the time domain capacitors and inductors need to have zero initial condition or else they are technically not linear components (for example, a zero input can give you a nonzero output, which linear systems by definition can't do; and yes, I know that you could model the initial condition as a separate dependent source). Here it looks like you need to assume the initial condition to be a certain nonzero value in order for the component to be linear in the phasor domain.
It's interesting that in the s-domain, you assume that the initial state is zero in order to get linear s-domain components (or alternatively you model the initial condition as a separate independent source). See here for reference.
Best Answer
The core of your question is
Recall that the exponential notation is for the stationary case. In other words, it assumes that the voltage oscillation has been applied for an infinite amount of time prior to t=0. Practically this assumption is satisfied when any transient effects, after starting for instance with v(0)=V and I(0)=0, have died out well before t=0, and only a pure and constant sinusoidal oscillation remains at t>0.
You can also cut into t=0, and start a pure and constant sinusoidal oscillation by applying the appropriate initial conditions as though the transients have settled. In your case this means starting with v(0) = V and i(0) as mathematically calculated in your question, and letting the system oscillate with v(t) after that.
This is a real physical concept. In linear systems, transients can be entirely eliminated by initializing the system with appropriate initial conditions, as you have done.