A time-domain representation for ultrabroadband optical pulses that includes chirp, stretching, and absolute phase is presented. This is accomplished by holomorphic Fourier transform of a class of spectra that include spectral phase up to second order, resulting in derivatives of the Faddeeva function w(ζ). Relating the real part of this pulse to physical quantities, its shape is investigated. The derivative order η is shown to dictate relative spectral bandwidth. The absolute phase determines the mixing of more localized (Gaussian decay) and less localized (algebraic decay) components of w(ζ). Second order spectral phase accounts for linear dispersion and is shown to cause a sheering of the Wigner-Ville distribution of the pulse, akin to the linear chirp that would be seen in the quasi-monochromatic case. Along with η it determines the pulse length and number of oscillations. A correspondence with the slowly varying envelope approximation is derived for large order η, although the algebraic decay persists.