《复变函数论》试题(E)

《复变函数论》试题（E ）

Ⅰ. Cloze Tests （20102=⨯ Points ）

1. If n

n n i n n z ⎪⎭

⎫ ⎝⎛++-=211，then lim =+∞→n n z . 2. If C denotes the circle centered at 0z and n is an integer ，then )

(1210=-⎰C n dz z z i π. 3. The radius of the power series ∑∞=+12)1(n n z n

is .

4. The singular points of the function 1cos )(2+=

z z z f are . 5. 0 ,sin s Re 2=⎪⎭

⎫ ⎝⎛n z z , where n is a positive integer. 6. =z e dz

d z 2sin . 7. Th

e main argument and the modulus o

f the number i +1 are .

8. The square roots of )0(>A Ai are .

9. The definition of z cos is .

10. Log )22(i += .

Ⅱ. True or False Questions (1553=⨯ Points)

1. If a function f is differentiable at a point 0z ，then it is continuous at 0z .（ ）

2. If a point 0z is a zero of order n of f ，then 0z is a pole of order n of f /1.（ ）

3. There is a non-constant entire function which maps the plane into the disk 1000||

4. A function f is differentiable at a point 000iy x z += if and only if whose real and imaginary parts are differentiable at ),(00y x and the Cauchy Riemann conditions hold there.（ ）

5. If a function f is continuous on the plane and

=⎰C dz z f )(0 for every simple

closed contour C , then it is an entire function. ( ) Ⅲ. Computations (3557=⨯ Points)

1. Find the integral ⎰+C z

dz z e 1

2, where C is the circle 7||=z .

2. Find the value of ⎰⎰==+-+2

35121)1(sin z z z z dz z dz z z e . 3. Let )

2)(1(1)(--=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

⎰-++=765)(2，where {}4|:|==z z C ，find )1(i f +'. 5. Given )0(2:,2)(πθθ≤≤=+=i e z C z z z f ，find dz z f C ⎰)(. Ⅳ. Proving (30310=⨯ Points)

1. Show that 020

914lim 242=++-⎰+∞→R C R dz z z z , where R C is the circle centered at 0 with radius R .

2. Suppose that f is an entire function and there is a constant M and a positive

integer m such that )(|||)(|C ∈∀≤z z M z f m . Prove that

m m z a z a z a z f +++= 221)(

for some constants 1a , m a a ,,2 and all z in the plane.

3. Show that the equation 01438=-+-z z z has just three roots in the unite disk.