Solutions to some problems from HW4
Math 5522H
III.6.19. Let f O(U ) C 2 (U ). Prove that f O(U ). If f O(U ) C n (U ), prove that f is n-times
dierentiabe (in C-sense).
Solution. First of all, since f C 2 (U ), f is dierentiable in the real sense. Let f
Solutions to some problems from HW10
Math 5522H
VIII.5.2. Let f be a non-constant holomorphic function in a domain D and let g be a non-constant holomorphic function in a doman D f (D). Show that the multiplicity of the function g f at a point z0 D
is the
Solutions to some problems from HW7
Math 5522H
V.8.21. Let = 1 +2 +3 +4 where 1 (t) = 2i+eit for /2 t 7/2, 2 (t) = eit for /2 t 9/2,
3 (t) = 2i + eit for /2 t 3/2, and 4 (t) = 3eit for /2 t /2. Calculate (z 4 + 4z 2 )1 ez dz .
z
Solution. The function f (
Solutions to Midterm 1 problems
1. What is the set dened by the equation
2z 3i
z +5
Math 5522H
= r, r > 0, r = 2? What if r = 2?
z 3
Solution. This set is the preimage of the circle cfw_w : |w| = 2 under the Mbius transformation (z ) = 2z+5 i ,
o
and so,
Midterm 1
Math 5522H
2z 3i
z +5
15%
= r, r > 0, r = 2? (Dont nd it exactly, just
say what it is a square, a parabola, .? And it would be good to avoid computations.) What if r = 2?
15%
2. If a set S C is path-connected, prove that it is connected.
15%
3.
Solutions to some problems from HW8
Math 5522H
V.8.58. Let f (z ) = (z z1 )(z z2 ) . . . (z zr ), where zi are distinct complex numbers, and let D be a domain
with the following property: for any 1 i [r/2], the points z2i1 and z2i are contained in a conne
Solutions to some problems from HW6
Math 5522H
V.8.6. Let and be closed, piecewise smooth paths in C with the same initial point. Demonstrate that
n(, z ) = n(, z ) and n( + , z ) = n(, z ) + n(, z ) for any z C \ (| + |).
dw
dw
dw
1
dw
Solution. n(, z )
Solutions to some problems from HW3
Math 5522H
III.6.32. Let z, w C \ (2N + 1) . Show that tan w = tan z i w = z + k for some k Z.
2
iz
iz
e
e
e2iz 1
1 eiz eiz
sin z
= eiz 2ieiz =
= i 2iz
, so tan is the composition of
Solution. We have tan z =
+
cos z
i
Solutions to some problems from HW2
Math 5522H
I.4.30. Conrm that z z = z + for all z, , C (where z w is understood in the sense of principal
values). Give an example where (z ) = z .
Solution. We have z z = e Log z e Log z = e Log z+ Log z = e(+) Log z =
Solutions to some problems from HW1
Math 5522H
I.4.4. Establish the parallelogram law: for any z, w C, |z + w|2 + |z w|2 = 2|z |2 + 2|w|2 .
Solution.
|z + w|2 + |z w|2 = (z + w)(z + w) + (z w)(z w) = (z + w)( + w) + (z w)( w)
z
z
= z z + z w + wz + ww + z
Midterm 2
20%
1. Let (t) = teit , t [0, ].
(a) How is
(b) Find
20%
20%
z
ez dz formally dened?
e dz .
2. Prove Moreras theorem: if f C (U ) is such that
in U , then f O(U ).
3. Find
20%
Math 5522H
sin z
dz , where =
z (z 1)
0
R
f (z ) dz = 0 for any recta