→ rachel5566 :謝謝您 03/25 01:10
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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