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The closed-form Green's functions (CFGF), derived for the vector and scalar potentials in planar multilayer media, have been revisited to clarify some issues and misunderstandings on the derivation of these Green's functions. In addition, the range of validity of these Green's functions is assessed with and without explicit evaluation of the surface wave contributions. As it is well-known, the derivation of the CFGF begins with the approximation of the spectral-domain Green's functions by complex exponentials, and continues with applying the Sommerfeld identity to cast these approximated spectral-domain Green's functions into the space domain in closed forms. Questions and misunderstandings of this derivation, which have mainly originated from the approximation process of the spectral-domain Green's functions in terms of complex exponentials, can be categorized and discussed under the topics of: 1) branch-point contributions; 2) surface wave pole contributions; and 3) the accuracy of the obtained CFGF. When these issues are clarified, the region of validity of the CFGF so obtained may be defined better. Therefore, in this paper, these issues will be addressed first, and then their origins and possible remedies will be provided with solid analysis and numerical demonstrations.