1
MATH 249 Homework #1, due Jan. 19, 2012
Problem 1: Describe geometrically the sets of points in the complex plane
dened by the following relations:
a) |z z1 | = |z z2 | where z1 , z2 C
b) 1/z = z
c)
MATH 249 Homework #1, due Jan. 19, 2012
1
Problem 1: Describe geometrically the sets of points in the complex plane
dened by the following relations:
a) |z z1 | = |z z2 | where z1 , z2 C
b) 1/z = z
c)
1
MATH 249 Homework #4, due Feb. 9, 2012
Problem 1: Let be open, simply connected, and f holomorphic on .
Let be a simple closed curve (with positive orientation) in , and a, b two
points in the inter
MATH 249 Homework #2, due Jan. 26, 2012
1
Problem 1: Determine the radius of convergence of the power series
n
n=1 an z , when
a) an = (log n)2
b) an = n!
c) an =
n2
4n +3n
d) an = sin( n)
4
Solution.
1
MATH 249 Homework #3, due Feb. 2, 2012
Problem 1: Show that the function f (z ) = 1/z is analytic in C \ cfw_0, i.e.,
it has a convergent power series expansion around any point w C with
w = 0. For
1
MATH 249 Homework #7, due March 15, 2012
Problem 1: Show that, for any > 0, the function f (z ) = e1/z , dened on
the disc cfw_|z | < of radius centered at the origin, takes on every value of
C exc
MATH 249 Homework #8, due March 22, 2012
1
Problem 1: For the principle branch of the logarithm, and given z1 C,
nd all values of z2 C such that
log(z1 z2 ) = log z1 + log z2
Solution. By denition, th
MATH 249 Homework #6, due March 8, 2012
1
Problem 1: Show that the complex zeros of the function f (z ) = sin(z ) are
exactly the integers, and that they are each of order 1. Calculate the residue
of
MATH 249 Homework #9, due March 29, 2012
Problem 1: Let
u(x, y ) =
1+x
(1 + x)2 + g (y )
where g is a real-valued function with g (0) = 0.
a) Find g such that u is harmonic on the domain |z + 1| > 0.
MATH 249 Homework #5, due March 1, 2012
1
Problem 1: Find all entire functions f : C C such that f (0) = 1 and
f (2z ) = f (3z ) for all z C.
Solution. Only the constant function f (z ) = 1 satises th
1
MATH 249 Homework #10, due April 5, 2012
Problem 1: Let f be a meromorphic function on C, with nitely many
poles cfw_z1 , . . . , zK , which are all simple and do not lie in Z. Assume that
|f (z )|
MATH 249 Homework #2, due Jan. 26, 2012
1
Problem 1: Determine the radius of convergence of the power series
n
n=1 an z , when
a) an = (log n)2
b) an = n!
c) an =
n2
4n +3n
d) an = sin( n)
4
Problem 2
MATH 249 Homework #4, due Feb. 9, 2012
1
Problem 1: Let be open, simply connected, and f holomorphic on .
Let be a simple closed curve (with positive orientation) in , and a, b two
points in the inter
MATH 249 Homework #3, due Feb. 2, 2012
1
Problem 1: Show that the function f (z ) = 1/z is analytic in C \ cfw_0, i.e.,
it has a convergent power series expansion around any point w C with
w = 0. For
1
MATH 249 Homework #6, due March 8, 2012
Problem 1: Show that the complex zeros of the function f (z ) = sin(z ) are
exactly the integers, and that they are each of order 1. Calculate the residue
of
1
MATH 249 Homework #5, due March 1, 2012
Problem 1: Find all entire functions f : C C such that f (0) = 1 and
f (2z ) = f (3z ) for all z C.
Problem 2: Let fn : C be sequence of holomorphic functions
MATH 249 Homework #8, due March 22, 2012
1
Problem 1: For the principle branch of the logarithm, and given z1 C,
nd all values of z2 C such that
log(z1 z2 ) = log z1 + log z2
Problem 2: Find all solut
MATH 249 Homework #9, due March 29, 2012
Problem 1: Let
u(x, y ) =
1
1+x
(1 + x)2 + g (y )
where g is a real-valued function with g (0) = 0.
a) Find g such that u is harmonic on the domain |z + 1| > 0
MATH 249 Homework #7, due March 15, 2012
1
Problem 1: Show that, for any > 0, the function f (x) = e1/z , dened on
the disc cfw_|z | < of radius centered at the origin, takes on every value of
C exce
1
MATH 249 Homework #10, due April 5, 2012
Problem 1: Let f be a meromorphic function on C, with nitely many
poles cfw_z1 , . . . , zK , which are all simple and do not lie in Z. Assume that
|f (z )|