复变函数论(A )
答卷注意事项:
、学生必须用蓝色(或黑色)钢笔、圆珠笔或签字笔直接在试题卷上答题。
2、答卷前请将密封线内的项目填写清楚。
3、字迹要清楚、工整,不宜过大,以防试卷不够使用。
4、本卷共 4 大题,总分为100分。
Ⅰ. Cloze Tests (20102=? Points )
1. If n
n
n n i i z ??
?
??++??? ??-=1173,then
lim =+∞
→n n z .
If C denotes the circle centered at 0z positively oriented and n is a
positive integer ,then
)
(1
0=-?C n dz z z . The radius of convergence of
∑∞
=++1
3
)123(n n z n n
is .
The singular points of the function )
3(cos )(22+=z z z
z f are .
0 ,)ex p(s Re 2=??
?
??n z z , where n is a positive integer.
=)sin (3z e dz
d z
. The main argument and the modulus of the number i -1 are .
8. The square roots of i -1 are . 9. The definition of z e is . 10. Log )1(i -= .
Ⅱ. True or False Questions (1553=? Points)
1. If a function f is analytic at a point 0z ,then it
is differentiable at 0z .( )
2. If a point 0z is a pole of order k of f ,then 0z is a zero of order k of
f /1.( )
3. A bounded entire function must be a constant.( )
4. A function f is analytic a point 000iy x z += if and only if whose real and
imaginary parts are differentiable at ),(00y x .( )
5. If f is continuous on the plane and =+?C
dz z f z ))((cos 0 for every simple
closed path C , then z e z f z 4sin )(+ is an entire function. ( )
Ⅲ. Computations (3557=? Points)
1. Find
?=-+1||)2)(12(5z z z zdz
.
2. Find the value of ??==-+22812
2)
1(sin z z z z dz
z dz z z
e .
3. Let )
2)(1()(--=
z z z
z f ,find the Laurent expansion of
f on the annulus
{}1||0:<<=z z D .
4. Given λλλλd z z f C ?-++=3
45)(2,where {}3|:|==z z C ,find )1(i f +-'.
5. Given )
1)(1(sin 1)(2+-+=z z z
z f ,find )1),(Res()1),(Res(-+z f z f .
Ⅳ. Verifications (30310=? Points)
1. Show that if )(0)()(C z z f k ∈?≡, then (f is a polynomial of order k <.
.
2. Show that 012
79
7lim 242=+++?+∞
→R C R dz z z z , where R C
at 0 with radius R .
3. Show that the equation 012524=-+-z z z has just two roots in the
unite disk