Preface This book is one of the key textbooks for high education,which was approved by the State Education Commission of China and the course:Methods of Mathematical Physics has been approved as a famous course by the State Education Commission of China since 1998. This book is divided into three parts,Function Theory of One Complex Variable, Equations of Mathematical Physics and Wavelets and Their Applications. The first part has four chapters.The first three chapters are devoted to the extension of real analysis to complex analysis.Any extension is feasible if and only if it is subjected to the three principles of extension.The fourth chapter introduces Residue Theorem and its applications to evaluation of definite integrals and series. The second part has nine chapters.In the first chapter,mathematical models of physical phenomena are constructed.The basic methods for the solutions of linear partial differential equations are introduced in the following five chapters.Special functions and their asymptotic expansions and their applications to equations of mathematical physics are dealt with in Chapters 7 and 8.And finally,inverse scattering problems and nonlinear problems are studied in Chapter 9. Much emphasis is placed on basic methods for the solutions of linear systems- methods of linear decomposition and synthesis,especially methods of spectral decomposition and impulse decomposition.The physical idea on which these two methods are based is very simple,that is the law of independence of disturbances,and the corresponding mathematical idea is the orthogonal expansion,in particular,the eigenfunction expansion.In this part,the method of separation of variables and the method of integral transforms are combined into one method as the method of spectral decomposition.Temporal impulses,spatial impulses,spatio-temporal impulses,initial impulses and boundary impulses are treated in a unified way in the method of impulse decomposition,including the Duhamel and the Riemann methods.Generalized functions get a chapter to themselves because they are indispensable to methods of continuous spectral decomposition and impulse decomposition and are greatly involved in the modern physics education and research.Generalized functions are also introduced based on the three principles of extension. The second part also gives the quintessence of the hypergeometric theory in only two sections.The method of traveling waves is also a useful one,especially the picture of traveling waves involved.The inverse problem and the nonlinear problem were two Ｐｒｅｆａｃｅ Ｔｈｉｓｂｏｏｋｉｓｏｎｅｏｆｔｈｅｋｅｙｔｅｘｔｂｏｏｋｓｆｏｒｈｉｇｈｅｄｕｃａｔｉｏｎ，ｗｈｉｃｈｗａｓａｐｐｒｏｖｅｄｂｙｔｈｅ ＳｔａｔｅＥｄｕｃａｔｉｏｎＣｏｍｍｉｓｓｉｏｎｏｆＣｈｉｎａａｎｄｔｈｅｃｏｕｒｓｅ：ＭｅｔｈｏｄｓｏｆＭａｔｈｅｍａｔｉｃａｌＰｈｙｓｉｃｓ ｈａｓｂｅｅｎａｐｐｒｏｖｅｄａｓａｆａｍｏｕｓｃｏｕｒｓｅｂｙｔｈｅＳｔａｔｅＥｄｕｃａｔｉｏｎＣｏｍｍｉｓｓｉｏｎｏｆＣｈｉｎａｓｉｎｃｅ １９９８． Ｔｈｉｓｂｏｏｋｉｓｄｉｖｉｄｅｄｉｎｔｏｔｈｒｅｅｐａｒｔｓ，Ｆｕｎｃｔｉｏｎ ＴｈｅｏｒｙｏｆＯｎｅＣｏｍｐｌｅｘ Ｖａｒｉａｂｌｅ， ＥｑｕａｔｉｏｎｓｏｆＭａｔｈｅｍａｔｉｃａｌＰｈｙｓｉｃｓａｎｄＷａｖｅｌｅｔｓａｎｄＴｈｅｉｒＡｐｐｌｉｃａｔｉｏｎｓ． Ｔｈｅｆｉｒｓｔｐａｒｔｈａｓｆｏｕｒｃｈａｐｔｅｒｓ．Ｔｈｅｆｉｒｓｔｔｈｒｅｅｃｈａｐｔｅｒｓａｒｅｄｅｖｏｔｅｄｔｏｔｈｅｅｘｔｅｎｓｉｏｎ ｏｆｒｅａｌａｎａｌｙｓｉｓｔｏｃｏｍｐｌｅｘａｎａｌｙｓｉｓ．Ａｎｙｅｘｔｅｎｓｉｏｎｉｓｆｅａｓｉｂｌｅｉｆａｎｄｏｎｌｙｉｆｉｔｉｓｓｕｂｊｅｃｔｅｄ ｔｏｔｈｅｔｈｒｅｅｐｒｉｎｃｉｐｌｅｓｏｆｅｘｔｅｎｓｉｏｎ．ＴｈｅｆｏｕｒｔｈｃｈａｐｔｅｒｉｎｔｒｏｄｕｃｅｓＲｅｓｉｄｕｅＴｈｅｏｒｅｍａｎｄ ｉｔｓａｐｐｌｉｃａｔｉｏｎｓｔｏｅｖａｌｕａｔｉｏｎｏｆｄｅｆｉｎｉｔｅｉｎｔｅｇｒａｌｓａｎｄｓｅｒｉｅｓ． Ｔｈｅｓｅｃｏｎｄｐａｒｔｈａｓｎｉｎｅｃｈａｐｔｅｒｓ．Ｉｎｔｈｅｆｉｒｓｔｃｈａｐｔｅｒ，ｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｓｏｆ ｐｈｙｓｉｃａｌｐｈｅｎｏｍｅｎａａｒｅｃｏｎｓｔｒｕｃｔｅｄ．Ｔｈｅｂａｓｉｃｍｅｔｈｏｄｓｆｏｒｔｈｅｓｏｌｕｔｉｏｎｓｏｆｌｉｎｅａｒｐａｒｔｉａｌ ｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓａｒｅｉｎｔｒｏｄｕｃｅｄｉｎｔｈｅｆｏｌｌｏｗｉｎｇｆｉｖｅｃｈａｐｔｅｒｓ．Ｓｐｅｃｉａｌｆｕｎｃｔｉｏｎｓａｎｄ ｔｈｅｉｒａｓｙｍｐｔｏｔｉｃｅｘｐａｎｓｉｏｎｓａｎｄｔｈｅｉｒａｐｐｌｉｃａｔｉｏｎｓｔｏｅｑｕａｔｉｏｎｓｏｆｍａｔｈｅｍａｔｉｃａｌｐｈｙｓｉｃｓａｒｅ ｄｅａｌｔｗｉｔｈｉｎＣｈａｐｔｅｒｓ７ａｎｄ８．Ａｎｄｆｉｎａｌｌｙ，ｉｎｖｅｒｓｅｓｃａｔｔｅｒｉｎｇｐｒｏｂｌｅｍｓａｎｄｎｏｎｌｉｎｅａｒ ｐｒｏｂｌｅｍｓａｒｅｓｔｕｄｉｅｄｉｎＣｈａｐｔｅｒ９． Ｍｕｃｈｅｍｐｈａｓｉｓｉｓｐｌａｃｅｄｏｎ ｂａｓｉｃ ｍｅｔｈｏｄｓｆｏｒｔｈｅｓｏｌｕｔｉｏｎｓｏｆｌｉｎｅａｒｓｙｓｔｅｍｓ— ｍｅｔｈｏｄｓ ｏｆ ｌｉｎｅａｒ ｄｅｃｏｍｐｏｓｉｔｉｏｎ ａｎｄ ｓｙｎｔｈｅｓｉｓ， ｅｓｐｅｃｉａｌｌｙ ｍｅｔｈｏｄｓ ｏｆ ｓｐｅｃｔｒａｌ ｄｅｃｏｍｐｏｓｉｔｉｏｎａｎｄｉｍｐｕｌｓｅｄｅｃｏｍｐｏｓｉｔｉｏｎ．Ｔｈｅｐｈｙｓｉｃａｌｉｄｅａｏｎｗｈｉｃｈｔｈｅｓｅｔｗｏｍｅｔｈｏｄｓ ａｒｅｂａｓｅｄｉｓｖｅｒｙｓｉｍｐｌｅ，ｔｈａｔｉｓｔｈｅｌａｗ ｏｆｉｎｄｅｐｅｎｄｅｎｃｅｏｆｄｉｓｔｕｒｂａｎｃｅｓ，ａｎｄｔｈｅ ｃｏｒｒｅｓｐｏｎｄｉｎｇ ｍａｔｈｅｍａｔｉｃａｌ ｉｄｅａ ｉｓ ｔｈｅ ｏｒｔｈｏｇｏｎａｌ ｅｘｐａｎｓｉｏｎ，ｉｎ ｐａｒｔｉｃｕｌａｒ，ｔｈｅ ｅｉｇｅｎｆｕｎｃｔｉｏｎｅｘｐａｎｓｉｏｎ．Ｉｎｔｈｉｓｐａｒｔ，ｔｈｅ ｍｅｔｈｏｄｏｆｓｅｐａｒａｔｉｏｎ ｏｆｖａｒｉａｂｌｅｓａｎｄｔｈｅ ｍｅｔｈｏｄｏｆｉｎｔｅｇｒａｌｔｒａｎｓｆｏｒｍｓａｒｅｃｏｍｂｉｎｅｄｉｎｔｏｏｎｅ ｍｅｔｈｏｄａｓｔｈｅ ｍｅｔｈｏｄｏｆｓｐｅｃｔｒａｌ ｄｅｃｏｍｐｏｓｉｔｉｏｎ．Ｔｅｍｐｏｒａｌｉｍｐｕｌｓｅｓ，ｓｐａｔｉａｌｉｍｐｕｌｓｅｓ，ｓｐａｔｉｏｔｅｍｐｏｒａｌｉｍｐｕｌｓｅｓ，ｉｎｉｔｉａｌ ｉｍｐｕｌｓｅｓａｎｄｂｏｕｎｄａｒｙｉｍｐｕｌｓｅｓａｒｅｔｒｅａｔｅｄｉｎａｕｎｉｆｉｅｄ ｗａｙｉｎｔｈｅ ｍｅｔｈｏｄｏｆｉｍｐｕｌｓｅ ｄｅｃｏｍｐｏｓｉｔｉｏｎ，ｉｎｃｌｕｄｉｎｇｔｈｅＤｕｈａｍｅｌａｎｄｔｈｅＲｉｅｍａｎｎ ｍｅｔｈｏｄｓ．Ｇｅｎｅｒａｌｉｚｅｄｆｕｎｃｔｉｏｎｓ ｇｅｔａｃｈａｐｔｅｒｔｏｔｈｅｍｓｅｌｖｅｓｂｅｃａｕｓｅｔｈｅｙａｒｅｉｎｄｉｓｐｅｎｓａｂｌｅｔｏ ｍｅｔｈｏｄｓｏｆｃｏｎｔｉｎｕｏｕｓ ｓｐｅｃｔｒａｌｄｅｃｏｍｐｏｓｉｔｉｏｎａｎｄｉｍｐｕｌｓｅｄｅｃｏｍｐｏｓｉｔｉｏｎａｎｄａｒｅｇｒｅａｔｌｙｉｎｖｏｌｖｅｄｉｎｔｈｅｍｏｄｅｒｎ ｐｈｙｓｉｃｓｅｄｕｃａｔｉｏｎａｎｄｒｅｓｅａｒｃｈ．Ｇｅｎｅｒａｌｉｚｅｄｆｕｎｃｔｉｏｎｓａｒｅａｌｓｏｉｎｔｒｏｄｕｃｅｄｂａｓｅｄｏｎｔｈｅ ｔｈｒｅｅｐｒｉｎｃｉｐｌｅｓｏｆｅｘｔｅｎｓｉｏｎ． Ｔｈｅｓｅｃｏｎｄｐａｒｔａｌｓｏｇｉｖｅｓｔｈｅｑｕｉｎｔｅｓｓｅｎｃｅｏｆｔｈｅｈｙｐｅｒｇｅｏｍｅｔｒｉｃｔｈｅｏｒｙｉｎｏｎｌｙｔｗｏ ｓｅｃｔｉｏｎｓ．Ｔｈｅ ｍｅｔｈｏｄｏｆｔｒａｖｅｌｉｎｇ ｗａｖｅｓｉｓａｌｓｏａｕｓｅｆｕｌｏｎｅ，ｅｓｐｅｃｉａｌｌｙｔｈｅｐｉｃｔｕｒｅｏｆ ｔｒａｖｅｌｉｎｇ ｗａｖｅｓｉｎｖｏｌｖｅｄ．Ｔｈｅｉｎｖｅｒｓｅｐｒｏｂｌｅｍ ａｎｄｔｈｅ ｎｏｎｌｉｎｅａｒｐｒｏｂｌｅｍ ｗｅｒｅｔｗｏ