推 chiehfu :thanks!!! 05/09 12:13
※ 引述《chiehfu ( Hate)》之銘言：
: D = {z: |Re(z)| < 1, |Im(z)| < 1}
: suppose f(z) is a function analytic on D and such that f(z) = 0
: on the side of D: f(z) = 0 on Re(z)=1, -1<Im(z)<1
: Consider g(z) = f(z)f(iz)f(-z)f(-iz)
: prove that f is identically zero in D
: 請問這題該怎麼證明
: 謝謝....
Since |g| vanishes on the boundary of D,
by maximal modulus priciple, we know that g = 0 inside D.
g(z) = f(z)f(iz)f(-z)f(-iz) = 0 for any z in D.
At least one of f(z), f(iz), f(-z), f(-iz) vanishes
infinitely many times inside D. May assume f(z), and it vanishes at {a_n}.
But Bozalno-Weierstrass theorem says that it must have a limit point in D.
Now by identity theorem, f(z)=0
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◆ From: 219.71.210.134
※ 編輯: yusd24 來自: 219.71.210.134 (05/09 11:49)