复变函数第四章Ch4.4.ppt

11 Sect. 4.4 Laurent 级数 函数的Laurent展开方法 * -(1/(1+z))的导数展开成幂级数-(∑ )的导数 Th.4.7 Laurent级数 Laurent展开式 R2 R1 * 1 证明思路： R r . z . 如图： 由多连通区域上的柯西积分公式： 展开即得。 * 2 1. 直接展开法（公式法：求cn） 由五个初等函数的泰勒展开式进行四则运算、 代换、变换、逐项求导、求积分等方法去展开. 2. 间接展开法（常用方法） * 3 E.g.1 Solut. * 4 * 5 * 6 E.g.2 Solut. x y O 2 1 . . * 7 x y O 2 1 . . Solut. E.g.2 * 8 E.g.3 Solut. * 9 * 10 补充：解析函数唯一性定理及其应用 解析函数唯一性定理 若函数 f, g ? A(D)，且 f, g在D内 某一子区域（或一小弧段）上相等，则 f 和 g在D内恒等。 注：定理中的小弧段常取为直线段来应用. 例. 若u(x,y)+iv(x,y) ?A(D), u(z,0)+iv(z,0) ?A(D)，且在D内含有实轴的一段L: z=x上相等，则在D内 u(x,y)+iv(x,y) = u(z,0)+iv(z,0) 注：由上例，可把一些符合条件的解析函数u(x,y)+iv(x,y) 化为 z的一元函数 u(z,0)+iv(z,0)（即，令x=z, y=0即可). * 11 Matthieu Paul Hermann Laurent，1841-1908，法国数学家。 Born: 2 Sept 1841 in Echternach, LuxembourgDied: 19 Feb 1908 in Paris, France He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry. He also developed statistical and interpolation formulae for calculating actuarial tables. His textbooks and his teaching of mathematics were particularly important, probably more so than his research contributions. Laurent married Berthe Moutard in 1874. He was appointed as professor at the école Agronomique in Paris in 1889 and he was honoured in 1905 when he was made Chevalier de la Légion dHonneur. Let us end with one final comment. Particularly given the topics on which Laurent worked, it is easy to assume that Laurent series must be named after him. However this is not the case and Laurent series are in fact named after Pierre Laurent. That is, Despite his large body of works, Laurent series expansions for complex functions were not named after him, but after Pierre Alphonse Laurent. Hermann Laurents mother was Anne-Fran?oise Schrobilgen and his father was Auguste Laurent. Hermanns paternal grandparents were Jean Baptiste Laurent, who owned a small farm, and Marie-Jeanne Maistre. The second of Jean and Mar