This book has comprehensively generalized analytical functions， proposed a new concept of semi-analytical functions， conjugate analytical functions and corresponding theories， and explained the applicatioof this theory ielectromagnetic fields， fluid mechanics， elastic mechanics， geometric transformatioand other fields. The theory is relatively systematic and the content is novel. Because it provides a method for how to propose new concepts and how to apply new concepts， it is aimportant reference book for scientists engaged itheoretical research and applied development research. This book caalso be used as a learning material for complex variable functions iscience and engineering colleges.