《复变函数论》试题(A)

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。