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We study existence of analytic solutions of a second-order iterative functional differential equation $x^{\text{'}\text{'}}\left(z\right)={\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(z\right){\left(x^{\left[j\right]}\left(z\right)\right)}^t+G\left(z\right)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $\alpha ²y^{\text{'}\text{'}}\left(\alpha z\right)y^{\text{'}}\left(z\right)=\alpha y^{\text{'}}\left(\alpha z\right)y^{\text{'}\text{'}}\left(z\right)+\left[y^{\text{'}}\left(z\right)\right]³[{\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(y\left(z\right)\right){\left(y\left({\alpha }^{j}z\right)\right)}^t+G\left(y\left(z\right)\right)]$ invertible analytic solutions of the form $y\left(\alpha y^{-1}\left(z\right)\right)$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the...