課程名稱︰複變函數論
課程性質︰大三必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試時間︰10/25 2007 10:20~12:10 (有延長10min...)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
COMPLEX ANALYSIS EXAMINATION I 10/25 2007
1. (10%) Find a rational function f(z) so that all its zeros and poles are
as follow:
zero (multiplicity) : z=1 (1), z=0 (2), z=2 (3)
pole (multiplicity) : z=5 (4), z=∞ (2).
x
2. (15%) Show that the real-valued function u(x,y) = ------- is harmonic
x^2+y^2
and find a real-valued function v(x,y) so that the complex function
f(z) = u(x,y)+iv(x,y), z=x+iy, is an analytic function.
3. (10%) Show that
x^3 (-1)^n ‧ x^(2n+1)
arctan(x) = x - --- + ... + ------------------ + ...,
3 2n + 1
and
π 1 (-1)^n
-- = 1 - - + ... + -------- + ...
4 3 2n + 1
4. (5% "each")
(a) In what situation an infinite sequence
z1, ..., z_n, z_n in C(complex), |z_n|≦1, for all n
form a discrete set? Why?
_
(b) Suppose C is connected. Show that the closure C is also connected.
(c) Suppose X,Y 包含於 R^n are compact. Show that if f: X → Y is
continuous, 1-1 and onto, then f is a topological map.
5. (15%) Let X = {z in C(complex) | |z|=1}. Explain why if D 包含於 C is
an open set where log(z) can be defined as a single-valued function,
then X 不包含於也不等於 D.
6. (15% "altogether")
(a) Find a linear transformation thet carries the unit circle |z|=1
to the real line Im(z)=0 so that the inner part of the unit circle
is sent to the upper half plane {z | Im(z) > 0}
(b) Find a linear transformation thet carries the region
_
{z | |z|<1 and |z-√2|<1} to the first quadrant
{z | Re(z)>0 and Im(z)>0}.
7. (10%) Find a conformal mapping that carries the region
{z | Re(z)>0, 0<Im(z)<π}
to the region {z | |z|<1}.
8. (10% "altogether") Let γ be a linear transformation.
(a) Suppose that z1, z2, z3, z4 are distinct complex numbers and
ω_j=γ(z_j) for j = 1,2,3. Show that if the cross ratios
(z1, z2, z3, z4) = (ω1,ω2,ω3,ω4),
then ω4 = γ(z4).
(b) Show that if γ carries the unit circle |z|=1 to a circle
with center z0 and γ(0)=z1, γ(∞)=z2, then z0, z1, z2 lie on a
straight line.
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