The expressions of commonly used conformal projections and their analytical transformations were thoroughly discussed with the help of complex numbers theory. The mathematical models of the forward and inverse solutions of Gauss, Mercator and conformal conic projections by complex numbers were given. Based on these models, the expressions of analytical transformations between Gauss, Mercator and conformal conic projections by complex numbers were systematically derived. These formulas by complex numbers are in the symbolic form including the first eccentricity of the reference ellipsoid, so they can solve the transformation problems when different reference ellipsoids are used. Compared with traditional transformation formulas in the real number domain, they have more concise structure and stricter theory basis.%借助复变函数理论讨论了常用等角投影及其解析变换的复变函数表示；给出了高斯投影、墨卡托投影和等角圆锥投影正反解的复变函数表示模型；在此基础上系统地推导出了高斯投影、墨卡托投影和等角圆锥投影间解析变换的复变函数表达式.这些复数变换公式是含参考椭球第一偏心率的符号形式,可解决不同参考椭球下的变换问题.与传统的实数变换公式相比,其结构更为简单、理论更为严密.
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