文档介绍：复变函数论（A）
I . Cloze Tests ( 2x 10 = 20 Points)
1.
If z
+ /[ 1 — I , then
2.
If C denotes the circle centered at z0 positively oriented and n is a
r 1
positive integer, then dz =
C(z — z°)
The radius of convergence of + 2" + l)z〃 is .
n=\
COS? z
The singular points of the function /(z) = are .
」 z(z +3)
fexp@) ) ,
Re si —云一,0 I -, where n is a positive integer.
— (ez sin3 z) = .
dz
The main argument and the modulus of the number 1 - i are .
The square roots of 1-z are .
The definition of ez is .
Log (1 - 0 =.
II. True or False Questions (3x5 = 15 Points)
If a function f is analytic at a point z0 , then it is differentiable at z0.
()
If a point z0 is a pole of order k ot f , then z0 is a zero of order k of
1//.(
A bounded entire function must be a constant.( )
A function f is analytic a point z0 = x0 + iyQ if and only if whose real and
imaginary parts are differentiable at (x0,y0).( )
If f is continuous on the plane and J (cos z + f(z))也=0 for every simple
c
closed path C , then /(z) + ez sin4 z is an entire function.( )
5zdz
III. Computations (7x5 = 35 Points)
侦=1 (2z + l)(z —2)
Find
c x i r f Wsin^z r z'dz
Find the value or ；——az + .
J, z2 J9(l-z)2
|z|=l |z|=2 \ '
Z r
Let /(z) = , find the Laurent expansion of J on the annulus
(z-l)(z-2)
= {z: 0 <| z |v 1}.
Given f(z) = f" + 42 * 3日人, where C = {z:|z|=3}, find /f(-l + 0.
]+ sin 之 z
Given f(z)= ~ , find Res(f (z),l) + Res(f (z),-l).
(z —l)(z + l)
IV・ Verifications (10x3 = 30 Points)
Show that if 了成)(z) = 0(Vz g C), then /(z)
is a polynomial of order < k .
. r 7z2 +9
Show that lim — dz = 0, where CR is the circle centered
g用日 Z4+7Z2+12 穴
at 0 with radius R .
Show that the equation z4 -5z2 +2z-l = 0 has just two roots in the
unite disk
复变函数论（B）
I . Cloze Tests ( 2x10 = 20 Points)
If C denotes the circle centered at z0 positively oriented and n is a
r 1
positive integer, then dz = .
c(z — z0)
The radius of the power series ^(n2 + l