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In gigacycle fatigue, crack initiation and growth most often occurs frominternal defects in the material including holes and inclusions. Occasionally a surfacedefect of hemi-spherical shape is also encountered. In order to attempt to understandthe stresses near these imperfections and the stress intensity factors for cracks initiatingfrom them, some elastic stress formulae will be developed here. For the inclusionsmismatches in elastic properties and sizes will be treated for realistic examination oftheir effects. It is hoped that convenient availability of such formulae may enhance anunderstanding of gigacycle fatigue initiation and crack growth.