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

f（z）在C中的任何地方均不可微分

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

4页
7
2个
3
2页

（i）有一个全纯函数f：C！ C（Re（f（z））= x2

（ii）如果g：C！ C在C上是全纯的，然后f（z）= jzj
2个
g（z）是

（iii）C上有一个全纯函数，使得f（
n + 1
n）= 0

（iv）连续函数f：Cnf0g！如果Cnf0g C是全纯的

[R

（i）
（ii）
（iii）
（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 di erentiable and holomorphic at i 2 C
 f(z) is di erentiable at 0 2 C but is not holomorphic at 0 2 C
 f(z) is di erentiable at i 2 C but is not holomorphic anywhere
in C
 f(z) is not di erentiable 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?
 4 p7
  p7
 2
  p3
 2 p5
 None of the others

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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