因为你许多条件没提到......我稍微补一下 首先 f is a "meromorphic function" on C (complex plane) . 它的定义如下: {a_0,a_1,a_2,...} is a set of distinct points that has no limit points in C. (i) the function f is holomorphic in C-{ a_0,a_1,a_2,...},and (ii) f has poles at the points { a_0,a_1,a_2,...}. 当 f 满足以下2个条件 (i) Every pole at finite is simple . (ii) We can choose { C_n } a series of concentric circls ( radius Rn ) about the oringin so that (1) the open disk D_n includes a_0,a_1,a_2...,a_n but no other poles . (2) Given any ε>0 , there exists a positive integer N such that |f(z)| <εRn for all |z| = Rn and n≧N 时,f(z)可以被展开成 ∞ ┌ 1 1 ┐ f(z) = f(0) + Σ (b_n) │ ──── + ─── │ n=1 └ z-(a_n) a_n ┘ where b_n = Res (f , a_n) 因为每个 pole 都是 "simple" 所以展开也相对简单 就如它的名称 pole expansion of Meromophic function (当pole不是simple, 条件(2)也要跟着改 , 展开也复杂些) 证明中为了简化 它假设 0<|a_0|<|a_1|<‧‧‧‧‧<|a_n|<‧‧‧‧ 任取 z (只要不是 0 或 f 的 poles 就好) 令 f(w)(w - a_n) g(w) = ──────── w (w-z) 积分路径为 C_n 计算 I_n =1/(2πi)* ∮g(w)dw 也就是算 C_n 里所有 Poles 的 residue 加总 g(w) 这函数 很直接就看出它有哪些poles (都是simple) {a_0,a_1,a_2,...}∪{0,z} (因为考虑到n趋近无穷,所以z要放进去) n I_n=Σ Res(g , a_m) + Res(g , 0) + Res(g , z) m=1 ( Res (g , a_m) = lim g(z)(z-a_m) z→a_m f(w)(w - a_m) b_m = lim ──────── = ─────── ) w→a_m w (w-z) a_m (a_m-z) ------------------------------------------------------ . . . 後续证明略,证明可以参考 Arfken, Weber - Mathematical Methods for Physicists pole expansion of Meromophic function ch7. 後面也有关於你问题 2 的说明... 问题条件不清楚就不好回答啊.....@@ --

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