Abstract: In this paper, an analytical method conformally transforming the semi-infinite domain of an arbitrary shape curve boundary into the one of a straight boundary line is given. By means of the calculative examples, it is proved that by making use of this method, the mapping accuracy is high and the calculation is simple. Thus the paper lays a foundation for the calculation of the stress and deflection of the tooth-shape machine elements which are commonly used in engineering.

Key words: the semi-infinite domain of atraight boundary lone, the semi-infinite domain of arbitrary shape curve boundary, conformal mapping