Math 4020 - Solutions of Assignment 1 - Winter 2011.
1. Let w be a nonzero complex number, and n be a positive integer. Present a
complete list of nth roots of w. Prove the following three facts about your list:
(i) Every element of your list is an nth ro
Math 4020/5020 - Solutions of Assignment 3 - Winter 2012.
1. Let , 1 and 2 be piecewise smooth curves. Let f : C C be continuous on
the curves , 1 and 2 . Prove that
(i)
(ii)
f (z )dz =
1 +2
f (z )dz =
f (z )dz.
1
f (z )dz +
2
f (z )dz .
(iii) If is a cl
Math 4020 - Solutions of Assignment 2 - Winter 2012.
1. Use the - denition of continuity to prove that the function f (z ) = z + z 3
is continuous on C. Find the largest subset of C on which f is dierentiable.
Reference any theorem that you use.
Solution:
Math 4020/5020 - Solution of Assignment 4 - Winter 2012.
1. Compute the following integrals.
2
(i)
ez z 3
dz ,
z +i
where is a piecewise smooth simple closed curve in the upper
half-plane oriented positively.
3
(ii)
z 2 ez
dz ,
z 2 +1
where is a piecewi
Math 4020 - Solution of Assignment 5 - Winter 2012.
1. Let an (z z0 )n be a power series of radius of convergence R > 0. Prove
n=0
that the power series converges uniformly on br (z0 ) for every r < R.
Solution: Fix r < R. We need to show that for every >
Math 4020 - Solutions of Assignment 6 - Winter 2012.
1. Let f (x + iy ) = u(x, y ) + iv (x, y ) be an entire function. Assume that u has an
upper bound in the xy -plane (i.e. there exists M R such that u(x, y ) M
for every (x, y ) R2 ). Prove that u must