因為你許多條件沒提到......我稍微補一下 首先 f is a "meromorphic function" on C (complex plane) . 它的定義如下: {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,...}. 當 f 滿足以下2個條件 (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 時,f(z)可以被展開成 ∞ ┌ 1 1 ┐ f(z) = f(0) + Σ (b_n) │ ──── + ─── │ n=1 └ z-(a_n) a_n ┘ where b_n = Res (f , a_n) 因為每個 pole 都是 "simple" 所以展開也相對簡單 就如它的名稱 pole expansion of Meromophic function (當pole不是simple, 條件(2)也要跟著改 , 展開也複雜些) 證明中為了簡化 它假設 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) 積分路徑為 C_n 計算 I_n =1/(2πi)* ∮g(w)dw 也就是算 C_n 裡所有 Poles 的 residue 加總 g(w) 這函數 很直接就看出它有哪些poles (都是simple) {a_0,a_1,a_2,...}∪{0,z} (因為考慮到n趨近無窮,所以z要放進去) 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) ------------------------------------------------------ . . . 後續證明略,證明可以參考 Arfken, Weber - Mathematical Methods for Physicists pole expansion of Meromophic function ch7. 後面也有關於你問題 2 的說明... 問題條件不清楚就不好回答啊.....@@ --

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