An exact hypergeometric implicit solution of the I D steady-state heat conduction problem for a straight fin of constant cross-section is used to calculate the dependence of the main dimensionless fin parameters, specifically, base thermal conductance G and thermo-geometrical fin parameter N on T-e, the ratio of the fin tip to fin base temperature excesses. The straight plate fin (SPF) and cylindrical pin fin (CPF) with an insulated tip (INT) and non-insulated tip (NINT) are optimized. The local heat transfer coefficient (HTC) is assumed to vary as power function of the local fin excess temperature with arbitrary value of exponent n in the range of -0.5 <= n <= 5. Every curve from G vs T-c set at given n for a fin with an INT is shown to have a single global maximum G = G(opt)* at T-c = T-opt* and corresponding N = N-opt* i.e. the main optimum parameters depend only on exponent n. Every curve from G vs T, set for a fin with a NINT depends, in addition, on the complex fin tip parameter B,,,. These curves have the local maximum and minimum points. As B,, increases these points approach each other and at B-omega = B** merge. The corresponding curve G vs T, has the only inflection point. The main optimum parameters of a fin with an INT and inflection point parameters of a fin with a NINT are approximated by general homographic function of n. Each main optimum parameter of a fin with a NINT is expressed as a product of the corresponding parameter of this fin with an INT and a correction factor approximated by the generalized closed-form formula, The results of the study are presented in form of dimensionless explicit relations, tables and plots which are well suited for the thermal design of optimum fins. (c) 2007 Elsevier Ltd. All rights reserved.