2.

f(z) = z + jzj
2个
+ iz

f（z）在0 2 C为全纯
f（z）在i 2 C时是可微且全纯的
f(z) 在 0 2 C 时可微，但在 0 2 C 时不是全纯的
f(z) 在 i 2 C 处可微，但在任何地方都不是全纯的

f(z) 在 C 中的任何地方都不可微

3.

; 0 t 2，对设备进行参数设置

ž

1个
z2 + 3z + 1
dz

Z 2
0
1个
3 + 2吨
dt

(i) 有一个全纯函数 f : C ！ C 与 Re(f(z)) = x2
.
(ii) 如果 g : C ！ C 在 C 上是全纯的，那么 f(z) = jzj
2个
g（z）是

(iii) 在 C 上存在一个全纯函数使得 f(
n + 1
) = 0

(iv) 连续函数 f : Cnf0g !如果Cnf0g C是全纯的

[R

（i）
(二)
(三)
（iv）

2.
Which one of the following statements about the complex di erentia-
bility of the function
f(z) = z + jzj
2
+ iz
is true?
 f(z) is holomorphic at 0 2 C
 f(z) is differentiable and holomorphic at i 2 C
 f(z) is differentiable at 0 2 C but is not holomorphic at 0 2 C
 f(z) is differentiable at i 2 C but is not holomorphic anywhere
in C
 f(z) is not differentiable anywhere in C
 None of the others

3.
Let be the contour (t) = eit
; 0  t  2, parametrising the unit
circle. By evaluating the contour integral
Z

1
z2 + 3z + 1
dz
in two di erent ways, the real integral
Z 2
0
1
3 + 2 cos t
dt
is seen to be equal to which one of the following values?

Consider the following statements:
(i) There is a holomorphic function f : C ! C with Re(f(z)) = x2
.
(ii) If g : C ! C is holomorphic on C then f(z) = jzj
2
g(z) is di eren-
tiable at z = 0 and at each zero of g.
(iii) There exists a holomorphic function on C such that f(
n+1
n ) = 0
for n = 1; 2; : : : and f(0) = 1.
(iv) A continuous function f : Cnf0g ! C is holomorphic on Cnf0g if
and only if
R
f(z) dz = 0 for every closed contour in Cnf0g
Which one of the statements holds true:
 (i)
 (ii)
 (iii)
 (iv)
 None of the others

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