Ⅱ. True or False Questions
1. (3 points) If a function f is differentiable at a point z0，then it is continuous at
z0.（ ）
2. (3 points) If a point z0 is a pole of order m of f，then z0 is a zero of order
m of 1/f.（ ）
3. (3 points) An entire function which maps the plane into the unite disk must be a
constant.（ ）
4. (3 points) A function f is differentiable at a point z0=x0+iy0 if and only if
whose real and imaginary parts are differentiable at (x0,y0) and the Cauchy Riemann conditions hold there.（ ）
5. (3 points) If a function f is continuous on the plane and
C
∫f(z)dz=0 for every
simple closed contour C, then f(z)sinz is an entire function. ( ) 6. If a function f is continuous at a point z0，then it is differentiable at z0（. ） 7. If a point z0 is a pole of order m of f，then there is analytic function ϕ at
z0 with ϕ(z0)≠0 such that f(z)=of z0.（ ）
8. An entire function which is identically zero on the real axis must be zero.（ ） 9. A function f is differentiable on a domain D if and only if whose real and
imaginary parts are differentiable on D and the Cauchy Riemann conditions hold on D.（ ）
10. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
ϕ(z)
(z−z0)
m
on some deleted neighborhood
closed contour C, then f(z)=0 for all z. ( )
11. If a function f is analytic at a point z0，then it is differentiable at z0.（ ） 12. If a point z0 is a pole of order k of f，then z0 is a zero of order k of
1/f.（ ）
13. A bounded entire function must be a constant.（ ）
14. A function f is analytic a point z0=x0+iy0 if and only if whose real and
1
imaginary parts are differentiable at (x0,y0).（ ）
15. If f is continuous on the plane and ∫(cosz+f(z))dz=0 for every simple
C
closed path C, then f(z)+ezsin4z is an entire function. ( )
16. If a function f is differentiable at a point z0，then it is continuous at z0（. ） 17. If a point z0 is a zero of order n of f，then z0 is a pole of order n of
1/f.（ ）
18. There is a non-constant entire function which maps the plane into the disk
|z|<1000.（ ）
19. A function f is differentiable at a point z0=x0+iy0 if and only if whose real
and imaginary parts are differentiable at (x0,y0) and the Cauchy Riemann conditions hold there.（ ）
20. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
closed contour C, then it is an entire function. ( ) 21. For all complex numbersz, sin2z+cos2z=1.（ ）
22. If a point z0 is a zero of order n of f，then z0 is a pole of order 2n of
1/f2.（ ）
23. There is a non-constant entire function which maps the complex plane into the
disk |z|<1.（ ）
24. A function f is differentiable at a point z0=x0+iy0 if and only if whose real
and imaginary parts are differentiable at (x0,y0) and the Cauchy Riemann conditions hold there.（ ）
25. If a function f is continuous on the plane and ∫(sinz+f(z)dz=0 for every
C
simple closed path C, then f(z)ezit is an entire function. ( )
26. If a function f is continuous at a point z0，then it is differentiable at z0（. ） 27. If a point z0 is a pole of order m of f，then there is analytic function ϕ at
z0 with ϕ(z0)≠0 such that f(z)=
ϕ(z)
(z−z0)
m
on some deleted neighborhood
2
of z0.（ ）
28. An entire function which is identically zero on the real axis must be zero.（ ） 29. A function f is differentiable on a domain D if and only if whose real and
imaginary parts are differentiable on D and the Cauchy Riemann conditions hold on D.（ ）
30. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
closed contour C, then f(z)=0 for all z. ( )
31. If a function f is continuous at a point z0，then it is differentiable at z0（. ） 32. If a point z0 is a pole of order m of f，then there is a function ϕ that is
analytic at z0 with ϕ(z0)≠0 such that f(z)=neighborhood of z0.（ ）
33. An entire function which is identically zero on a line segment must be identically
zero.（ ）
34. A function f is differentiable on open set D if and only if whose real and
imaginary parts are differentiable on D and the Cauchy Riemann conditions hold on D.（ ）
35. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
ϕ(z)
(z−z0)
m
on some deleted
closed path C, then f(z)=0 for all z. ( )
36. If a function f is differentiable at a point z0，then it is analytic at z0.（ ） 37. If a point z0 is a pole of order k of f，then z0 is a zero of order k of
1/f.（ ）
38. A bounded entire function must be a constant.（ ）
39. If a function f is differentiable at a point z0，then it is continuous at z0（. ） 40. If a point z0 is a zero of order n of f，then z0 is a pole of order n of
1/f.（ ）
41. There is a non-constant entire function which maps the plane into the disk
|z|<1000.（ ）
3
42. A function f is differentiable at a point z0=x0+iy0 if and only if whose real
and imaginary parts are differentiable at (x0,y0) and the Cauchy Riemann conditions hold there.（ ）
43. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
closed contour C, then it is an entire function. ( )
44. A function f is analytic a point z0=x0+iy0 if and only if whose real and
imaginary parts are differentiable and the Cauchy Riemann conditions hold in a neighborhood of(x0,y0).（ ）
45. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
closed contour C, then f(z)+ezsinz is an entire function. ( )
46. If a function f is continuous at a point z0，then it is differentiable at z0（. ） 47. If a point z0 is a pole of order m of f，then there is analytic function ϕ at
z0 with ϕ(z0)≠0 such that f(z)=of z0.（ ）
48. An entire function which is identically zero on the real axis must be zero.（ ） 49. A function f is differentiable on a domain D if and only if whose real and
imaginary parts are differentiable on D and the Cauchy Riemann conditions hold on D.（ ）
50. If a function f is continuous on the plane and ∫f(z)dz=0 for every simple
C
ϕ(z)
(z−z0)
m
on some deleted neighborhood
closed contour C, then f(z)=0 for all z. ( )
4
