[複變] analytic定義
f is analytic at a:(a is a point)
there exists d > 0 , s.t. f is differentiable on D(a;d)
f is analytic on S:(S is a set)
f is differentiable at all points on S
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Combine these two definition
Boas的書有提到
if f is analytic on S
代表存在一個open set U, S is contained in U
也就是說對於S的每個點存在一個disk,在這個disk可微
所以這些disk的聯集就是U了
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所以如果題目給個條件是:
f is analytic on F, F is closed
代表存在一個open cover 蓋住 F
而 f 在這個open cover 可微
問題來了
如果題目要你證明
"f is analytic on F, F is closed"
為什麼我們只需要證明 f 在 F 的每一點均可微
不需要證明存在一個open cover 在其上均可微
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03/24 00:36, , 1F
03/24 00:36, 1F