課程名稱︰複變函數論
課程性質︰大三必修
課程教師︰王藹農
開課學院：理學院
開課系所︰數學系
考試時間︰Nov. 16 2006 10:20~12:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Complex Variable
Nov. 16 2006
1. Let x^2+y^2+z^2=1 be the Riemann sphere. N=(0,0,1) be the north pole.
z=-1 be the stereographic projection to the complex plane. x=2/3 cuts a
small circle on the Riemann sphere. Its projection has center
(x,y,z)=(?,0,-1)
2. |z^2-1| < 2^2 is a lemniscate. w=f(z) is the conformal map onto the unit disc
_ ____
so that f(-z)=-f(z), f(z)=f(z). f(1+0i)=?
3. p=1/2 ,q=i/2, γ is a path from p to q. Length
|dz|
L(γ)= ∫ ──── (|z|<1). inf L(γ) =?
γ 1-|z|^2 γ
4.
dz _______ __
∫ ──── = ? At z=2, we choose √(1-2^2) = + √3 i
|z|=2 √(1-z^2) and √(1-z^2) is continous as we go around
|z|=2 counterclockwise.
5. △u=0 in the unit disc |z|<1, but its bondary value is not continous.
u(exp(iθ))=1 if 0<θ<π/4 , u(exp(iθ))=0 if π/4<θ<2π.
At z= -1/2, u=?
________
6. x<+√(1+y^2) is the region to the left of the right branch of a hyperbola.
_ ____
w=f(z) is the conformal map onto the half plane Re(w)<0, so that f(z)=f(z)
f(1)=0, f(0)=-1, f(-1)= ?
7. circles perpendicular to both |z|=1 and |z-(1/4)| = 1/4 have common points
z=? & ?
8. How many roots are there to the equation 1/3exp(z)=z in the unit disc
|z|<=1 ?
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