讲授要点 O Laurent展开 展开定理 举例 多值函数的 Laurent展开 0单值函数的孤立奇点 孤立奇点 孤立奇点的分类 。函数在无穷远处的奇异性 ③解析延拓 个例子 。解析延拓的概念
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions ùÇ: 1 LaurentÐm Ðm½n Þ~ õ¼êLaurentÐm 2 ü¼êáÛ: áÛ: áÛ:©a ¼ê3á?ÛÉ5 3 )Ûòÿ ~f )ÛòÿVg C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari anf iief 展开定理( Laurent) 设函数f()在以b为圆心的环 形区域B1<|z-b≤B2上单 值解析,则对于环域内的任 何z点,f(z)可以展开为 Laurent级数 f(2)=∑a(-b)B1<|2-b<B2 n=一 f() C是环域内绕内圆一周的任意一条闭合曲线 第八讲解析函数的 Laurent
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions Ðm½n(Laurent) ¼êf(z)3±b % /«R1 ≤ |z − b| ≤ R2þü )Û§ KéuS? Ûz:§f(z)±Ðm Laurent?ê f(z) = X ∞ n=−∞ an(z − b) n R1 < |z − b| < R2 an = 1 2π i I C f(ζ) (ζ − b) n+1dζ C´S7S ±?¿^4Ü C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari Anf taie con tinction 展开定理( Laurent) (要点) f()=∑a1(-yB2<|-M<R2 2πiJe(-b)n+1 dc 将环域的内外边界分别记为C1和C2,根据复连通 区域的Cach积分公式,对于环形区域内的任意 下面分别计算沿C1和C2的积分
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions Ðm½n(Laurent) (:) f(z)= X∞ n=−∞ an(z−b) n R1 <|z−b|<R2 an = 1 2π i I C f(ζ) (ζ−b) n+1 dζ òS >.©OPC1ÚC2§âEëÏ «CauchyÈ©úª§éu/«S?¿ :z§k f(z) = 1 2π i I C2 f(ζ) ζ−z dζ − 1 2π i I C1 f(ζ) ζ−z dζ e¡©OO÷C1ÚC2È© C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari Anf taie con tinction 展开定理( Laurent) (要点) f()=∑a1(-yB2<|-M<R2 2πiJe(-b)n+1 dc 将环堿的内外边界分别记为C1和C2,根据复连通 区域的 Cauchy积分公式,对于环形区域内的任意 点z,有 f() 1Cf() d 2πiJC1 面分别计算沿C1和C2的积分
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions Ðm½n(Laurent) (:) f(z)= X∞ n=−∞ an(z−b) n R1 <|z−b|<R2 an = 1 2π i I C f(ζ) (ζ−b) n+1 dζ òS >.©OPC1ÚC2§âEëÏ «CauchyÈ©úª§éu/«S?¿ :z§k f(z) = 1 2π i I C2 f(ζ) ζ−z dζ − 1 2π i I C1 f(ζ) ζ−z dζ e¡©OO÷C1ÚC2È© C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari Anf taie con tinction 展开定理( Laurent) (要点) f()=∑a1(-yB2<|-M<R2 2πiJe(-b)n+1 dc 将环堿的内外边界分别记为C1和C2,根据复连通 区域的 Cauchy积分公式,对于环形区域内的任意 点z,有 f(a 1 f( 1Cf() ds- 2Tti jcu 下面分别计算沿C1和C2的积分
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions Ðm½n(Laurent) (:) f(z)= X∞ n=−∞ an(z−b) n R1 <|z−b|<R2 an = 1 2π i I C f(ζ) (ζ−b) n+1 dζ òS >.©OPC1ÚC2§âEëÏ «CauchyÈ©úª§éu/«S?¿ :z§k f(z) = 1 2π i I C2 f(ζ) ζ−z dζ − 1 2π i I C1 f(ζ) ζ−z dζ e¡©OO÷C1ÚC2È© C. S. Wu 1lù )Û¼êLaurentÐm