Ⅳ. Proving
1. Show that the function is an entire function.
iy x e e z z f −−−=)2()(22. Show that if , then is a polynomial of order .
)(0)()(C z z f m ∈∀≡)(z f m <3. Show that 06
51lim 242=+++∫+∞→R C R dz z z z , where is the circle centered at with radius R C 0R .
4. Suppose that and f f are differentiable on a domain , prove that D A z f =)( for some constant A and all D z ∈.
5. Show that the equation has just two roots in the unite disk.
01424=−+−z z z 6. Show that 02316lim 242=+++∫+∞→R
C R dz z z z , where is the circle centered at with radius R C 0R .
7. Suppose that is differentiable and is a constant on a domain , prove
that for some constant f ||f D A z f =)(A and
all D z ∈. 8. Show that the equation has just three roots in the unite disk.
0127234=−++−z z z z 9. Show that if , then is a polynomial of order .
)(0)()(C z z f k ∈∀≡)(z f k <10. Show that 012
797lim 242=+++∫+∞→R C R dz z z z , where is the circle centered at with radius R C 0R .
11. Show that the equation has just two roots in the unite disk.
012524=−+−z z z 12. Show that 020
914lim 242=++−∫+∞→R C R dz z z z , where is the circle centered at with radius R C 0R .
13. Suppose that is an entire function and there is a constant f M and a positive integer such that . Prove that
m )(|||)(|C ∈∀≤z z M z f m
m m z a z a z a z f +++=L 221)(for some constants , and all in the plane.
1a m a a ,,2L z 14. Show that the equation has just three roots in the unite disk.
01438=−+−z z z 15. Show that if , then is a polynomial of order . )(0)()(C z z f m ∈∀≡)(z f m <
16. Show that 012
783lim 242=+++∫+∞→R C R dz z z z , where is the circle centered at with radius R C 0R .
17. Show that the equation has just two roots in the unite disk.
012524=−+−z z z 18. Show that 020975lim 242=++−∫+∞→R
C R dz z z z , where is the circle centered at with radius R C 0R .
19. Suppose that is an entire function and there is a constant f M and a positive
integer such that . Prove that
m )(|||)(|C ∈∀≤z z M z f m
m m z a z a z a z f +++=L 221)(for some constants , and all in the plane.
1a m a a ,,2K z 20. Show that the equation 1438+−=z z z has just three roots in the unite disk.
21. Show that 02316lim 242=+++∫+∞→R
C R dz z z z , where is the circle centered at with radius R C 0R .
22. Suppose that is differentiable and is a constant on a domain , prove
that for some constant f ||f D A z f =)(A and
all D z ∈. 23. Show that the equation has just three roots in the unite
disk.
0127234=−++−z z z z 24. Show that 0233lim 242=+++∫+∞→R
C R dz z z z , where is the circle centered at with radius R C 0R .
25. Show that if , then is a polynomial of order .
)(0)()(C z z f m ∈∀≡)(z f m <26. Show that 012
783lim 242=+++∫+∞→R C R dz z z z , where is the circle centered at with radius R C 0R .
27. Show that the equation has just two roots in the unite disk. 012524=−+−z z z。