Re: [理工] [工數] 98台聯大工數A
※ 引述《AAJJBurnett (叫我投手)》之銘言：
: Consider a branch f(z) of (z^2-1)^(1/2) that is analytic in
: the exterior of the unit circle , |z|>1.
: If f(根號2)=-1 ,find f(-i根號2).
單位圓內有條分支線喔!
iθ1+2kπi
令 z - 1 = r1 e
iθ2+2kπi
z + 1 = r2 e
-π < θ1 < π
-π < θ2 < π
i(θ1+θ2)/2+kπi
f(z) = √(r1r2) e k = 0 ,1
iθ1
√2 - 1 = r1 e
θ1 = 0
r1 = √2 - 1
iθ2
√2 + 1 = r2 e
θ2 = 0
r2 = √2 + 1
kπi
f(z) = 1 e , k = 0 , 1
= -1 (故 k = 1 )
i(-π)/2 + πi
f(-√2i) = √((√3)(√3)) e
= √3 i
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