数学物理方法Chapt 1 _6.pptxVIP

Contents of Chapt. 1:Complex Algebra (加减乘除,乘幂/开方,指数/取对数) Cauchy-Riemann Conditions (极限, 微分)Cauchy Integral Theorem Formula (回路积分) 2 sectionsTaylor and Laurent expansions (级数)Mapping(映射)Contents of Chapt. 2:Singularities Calculus of ResiduesChapt 1: Complex functionsshi.xq@sustc.edu.cnChapter 1: Complex functions 复函数P360 of EMMP1.6 Mapping Conformal Mapping共形映射、共形变换（保角变换）Picture from wikiChapt 1: Complex functionsshi.xq@sustc.edu.cnMapping 映射more geometric aspects of complex functionsFor real functions , plot y versus x within one planeFor complex functions z-plane to w-palne 示意图(u2 , v2)(x2 , y2)(x1 , y1)(u1 , v1)Chapt 1: Complex functionsshi.xq@sustc.edu.cn1.6 Mapping（转换或映射）As points in the z-plane transform (or are mapped) into points in the w-plane, lines or areas in the z-plane will be mapped into lines or areas in the w-plane.Our immediate purpose is to see how lines and areas map from the z-plane to the w-plane for a number of simple functions.几种映射：Translation （平移）Rotation （旋转和伸缩）Inversion （求逆和反转） Inversion of lines and circlesChapt 1: Complex functionsshi.xq@sustc.edu.cn1.6 Mapping1. Translation（平移）(u, v)(x, y)w is equal to z plus a constant, z0 = x0 + iy0representing a pure translation（平移变换）rChapt 1: Complex functionsshi.xq@sustc.edu.cn1.6 Mapping2. Rotation（旋转和伸缩）??For rotation, the polar representation is convenient thenorTwo things have occurred：the modulus r has been modified, either expanded or contracted, by the factor r0. the argument θ has been increased by the constant θ0. This represents a rotation of the complex variable through an angle θ0. 伸缩旋转Chapt 1: Complex functionsshi.xq@sustc.edu.cn1.6 MappingRotation旋转和伸缩r0 1 的情形For the special case of z0 = i, we have a pure rotation through π/2 radians.Chapt 1: Complex functionsshi.xq@sustc.edu.cn3. Inversion（求逆和反转）Again, using the polar form, we haveor(1, 0)r 1 的情形Chapt 1: Complex functionsshi.xq@sustc.edu.cn1.6 MappingInversion求逆和反转Again, two things have occurred: The interio