What is Complex amplitude?

To the sinusoid $x(t) = X\cos(\omega t + \varphi)$ one associates the complex signal $\underline{x}(t) = X\eu^{j(\omega t + \varphi)}$, so that $x = \Rea(\underline{x})$, and the complex amplitude

which carries the amplitude $X = \abs{\underline{X}}$ and the phase $\varphi = \arg\underline{X}$. Here $j$ denotes the imaginary unit, $j^2 = -1$, the letter $i$ being reserved for currents. Drawn as a vector of length $X$ at angle $\varphi$, $\underline{X}$ is a phasor (Fresnel vector).