This book has comprehensively generalized analytical functions, proposed a new concept of semi-analytical functions, conjugate analytical functions and corresponding theories, and explained the application of this theory in electromagnetic fields, fluid mechanics, elastic mechanics, geometric transformation and 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 an important reference book for scientists engaged in theoretical research and applied development research. This book can also be used as a learning material for complex variable functions in science and engineering colleges.