《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