《复变函数论》试题(D)

《复变函数论》试题（D ）

Ⅰ. Cloze Tests （20102=⨯ Points ）

1. If n n n n i i z ⎪⎭

⎫ ⎝⎛++⎪⎭⎫ ⎝⎛-=1153，then lim =+∞→n n z . 2. If C denotes the circle centered at 0z positively oriented and n is a positive integer ，then )(10=-⎰C

n dz z z . 3. The radius of the power series ∑∞=++13)12(n n z n n

is .

4. The singular points of the function )3(cos )(2+=

z z z z f are . 5. 0 ,)exp(s Re 2=⎪⎭

⎫ ⎝⎛n z z , where n is a positive integer. 6. =)sin (5z e dz

d z . 7. Th

e main argument and the modulus o

f the number i -1 are .

8. The square roots of 1+i 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 differentiable at a point 0z ，then it is analytic 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 and the Cauchy Riemann conditions hold in a neighborhood of ),(00y x .（ ）

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

=⎰C dz z f )(0 for every simple

closed contour C , then z e z f z sin )(+ is an entire function. ( )

Ⅲ. Computations (3557=⨯ Points)

1. Find ⎰=-+1||)2)(12(z z z zdz .

2. Find the value of ⎰⎰==-+2231

22)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

⎰-++=142)(2，where {}3|:|==z z C ，find )1(i f +-'. 5. Given )

1)(1(sin )(2+-=z z z z f ，find )1),(Res()1),(Res(-+z f z f . Ⅳ. Proving (30310=⨯ Points)

1. Show that if )(0)()(C z z f m ∈∀≡, then )(z f is a polynomial of order m <.

2. Show that 012

783lim 242=+++⎰+∞→R C R dz z z z , where R C is the circle centered at 0 with radius R .

3. Show that the equation 012524=-+-z z z has just two roots in the unite disk.