= Reactance phase convention
{title2=$v(t)=\operatorname{Re}(\widehat V e^{\mathrm i\omega t})$}
Use <complex amplitudes> only for signals at one common positive <angular frequency> $\omega$. Define the real terminal voltage and <electric current> by
$$
v(t)=\operatorname{Re}\left(\widehat V e^{\mathrm i\omega t}\right),\qquad
i(t)=\operatorname{Re}\left(\widehat I e^{\mathrm i\omega t}\right),\qquad
Z=\frac{\widehat V}{\widehat I}.
$$
The <complex amplitudes> here use peak magnitudes; both can instead use <root mean square> magnitudes without changing their ratio. Choose positive <electric current> entering the terminal assigned positive voltage, so $p=vi$ is absorbed <power>. For an ideal linear <inductor> and <capacitor>, their terminal laws $v_L=L\,di/dt$ and $i_C=C\,dv_C/dt$ give
$$
\widehat V_L=\mathrm i\omega L\widehat I,\qquad
\widehat I_C=\mathrm i\omega C\widehat V_C,\qquad
Z_L=\mathrm i\omega L,\qquad
Z_C=\frac1{\mathrm i\omega C}=-\frac{\mathrm i}{\omega C}.
$$
The <derivative> multiplies a <complex amplitude> by $\mathrm i\omega$: this single algebraic step explains both signs. Taking $i(t)=I_0\cos\omega t$ as a reference gives $v_L(t)=-\omega LI_0\sin\omega t$ and $v_C(t)=I_0\sin\omega t/(\omega C)$ for the sinusoidal response. Thus <inductor> voltage leads <electric current> by $\pi/2$, while <capacitor> voltage lags <electric current> by $\pi/2$. A constant offset or a free transient is outside this single-<frequency> representation. https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S1[Feynman, Leighton and Sands, II.22.1: complex sinusoidal representation and component impedances]{ref}.
With the alternative $e^{-\mathrm i\omega t}$ convention, the same real signals use conjugated <complex amplitudes> and $Z_L=-\mathrm i\omega L$, $Z_C=+\mathrm i/(\omega C)$. Their physical leads and lags stay the same; the sign of $\operatorname{Im}Z$ changes. Also, some textbooks call the positive magnitude $1/(\omega C)$ the capacitive reactance and write the series expression as $X_L-|X_C|$. This article consistently uses signed $X_C=-1/(\omega C)$ and hence $X=X_L+X_C$. https://openstax.org/books/university-physics-volume-2/pages/15-2-simple-ac-circuits[Ling, Moebs and Sanny, University Physics 2, §15.2: positive capacitive-reactance magnitude and quarter-cycle phase relations]{ref}.
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