HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 1
DUE MONDAY, SEPTEMBER 21, 2009
(1) (Bak and Newman, problem 1.5) Suppose P is a polynomial with real coecients. Show that P (z ) = 0 if and only if P (z ) = 0 (i.e., zeroes of real
polynomials come in com
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 5
DUE MONDAY, OCTOBER 19, 2009
Changes since rst version:
Added a missing r in problem (2), in the denition of g ().
Modied hint in problem (3).
In problem (5), consider regular polygons.
(1) Find the po
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 4
DUE MONDAY, OCTOBER 12, 2009
(1) Compute explicitly
z dz
R
where R is an arbitrary rectangle in C: from a to b on the x-axis, and from
c to d on the y -axis.
(2) (From Bak and Newman, problem 4.6)
(a) For
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 3
DUE MONDAY, OCTOBER 5, 2009
(1) Recall that the hyperbolic sine and hyperbolic cosine are dened by
(2)
(3)
(4)
(5)
(6)
ex ex
sinh(x) =
2
ex + ex
cosh(x) =
.
2
Find expressions for sinh(z ) and cosh(z ) in
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 2
DUE MONDAY, SEPTEMBER 28, 2009
(1) Prove the other direction of the radius of convergence theorem: in the power
series
ak z k , if |z | > R then the series diverges. (As usual, set 1/R =
lim sup |ak |1/k
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 6
OPTIONAL, NOT GRADED
(1) (Bak and Newman, problem 6.9) Find the maximum and minimum moduli
of z 2 z on the disk |z | 1.
(2) (Bak and Newman, problem 6.11) Derive the Fundamental Theorem of Algebra from th
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 7
DUE WEDNESDAY, NOVEMBER 4, 2009
This version:
added a missing hypothesis to 4(a) and 4(b);
claried the statement to prove in 4(c).
(1) (Bak and Newman, problem 9.5) Suppose that f and g have poles at z0
Honors Complex Variables, Math W4065
Mid-term exam
Due October 26, 2009, at 4:10 PM
This exam is open notes and open book: you may use any of your notes, the notes posted
online, and any of the three textbooks for this course. (Remember that all textbooks
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 11
DUE MONDAY, DECEMBER 14, 2009
This version (Dec. 8):
Corrected the denition of U : it should refer to maps to the unit disk, not
the whole complex plane.
Claried the complement of U in the last problem
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 10
DUE WEDNESDAY, DECEMBER 2, 2009
(1) (SteinShakarchi, 5.6.13) Show that the equation ez = z has innitely many
solutions in C. (You may use Hadamards theorem, which we stated but did
not prove.)
(2) Find a
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 8
DUE MONDAY, NOVEMBER 16, 2009
This version:
Claried that P and Q are polynomials in problem 2.
(1) (From Bak and Newman, problems 10.2 and . . . ) Evaluate the following
integrals.
(a) |z|=1 cot z dz
dz
HONORS COMPLEX VARIABLES, MATH W4065
PROBLEM SET 9
DUE WEDNESDAY, NOVEMBER 25, 2009
(1) (BakNewman, 9.7) Find the Laurent expansion for
1
(a) z4 +z2 around z = 0,
2
(2)
(3)
(4)
(5)
(b) exp(1/z ) around z = 0,
z 1
(c) z21 4 around z = 2.
1
Evaluate +=1 n4