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) Re(z ) = 3
d) Re(z ) c, where c R
e) Re(az + b) > 0, w
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) Re(z ) = 3
d) Re(z ) c, where c R
e) Re(az + b) > 0, w
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 interior of . Show that
1
2i
f ( a) f ( b )
f (z )
dz =
(z a
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.
a) Note that lim supn |an |1/n = 1 only if |an | depen
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 given w, what is its radius of convergence?
Solution. F
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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 except for 0 innitely many times.
Solution. For z = x + iy
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, the principle branch of the logarithm is given by
log z =
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 1/f (z ) at all z Z.
Solution. Since sin(z ) = (2i)1 (e
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.
b) Find the conjugate harmonic function of u.
Solution.
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 these assumptions.
Here are (at least) three possible way
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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 )| C/|z |2 for large |z |. In class, we have shown that
K
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: Let denote the Laplacian on R2 , i.e.,
=
2
2
+2
x2 y
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 interior of . Show that
1
2i
f ( a) f ( b )
f (z )
dz =
(z a
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 given w, what is its radius of convergence?
Problem 2:
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/f (z ) at all z Z.
Problem 2: Show that
e a
cos x
dx
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 that
converges uniformly to some f . (We have shown in
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 solutions of the equation
z
ee = 1
Problem 3: Let f be a hol
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.
b) Find the conjugate harmonic function of u.
Problem
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 except for 0 innitely many times.
Problem 2: Let C be an op
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 )| C/|z |2 for large |z |. In class, we have shown that
K