1个
z2 + 3z + 1
dz
有两种不同的方式，即真正的积分
Z 2
0
1个
3 + 2吨
dt
被视为等于以下哪个值
考虑以下语句：
(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
在z = 0且g的每个零处均成立。
(iii) 在 C 上存在一个全纯函数使得 f(
n + 1
) = 0
对于 n = 1; 2; : : : 和 f(0) = 1。
(iv) 连续函数 f : Cnf0g !如果Cnf0g C是全纯的
并且只有当
[R
对于Cnf0g中的每个闭合轮廓，f（z）dz = 0
以下哪一项陈述是正确的：
（i）
(二)
(三)
（iv）
其他都没有
2.
Which one of the following statements about the complex dierentia-
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 dierent 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 dieren-
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