Record of daily activities and homework, Math
618, Theory of Functions of a Complex Variable II,
Spring 2004
- Wednesday, January 21
- We worked on an exercise on the definition
of infinite products.
- Homework for Friday: Read section 8.1, pages
255-263.
- Friday, January 23
- We discussed the Weierstrass factorization theorem and the
representation of the sine function as an infinite product.
- Homework for Monday: Read section 8.2, pages
263-266, and do exercises 13 and 14 on page 276
(Chapter 8).
- Monday, January 26
- We discussed the extension to arbitrary planar domains of the
Weierstrass theorem about the existence of holomorphic functions
with prescribed zeroes.
- Homework for Wednesday: Read the first part of
section 8.3, pages 266-270, and do exercises 10 and 21 on
pages 276-277 (Chapter 8).
- Wednesday, January 28
- We discussed (i) variations on the proof of the
Weierstrass theorem and (ii) formulas for stereographic
projection and the spherical distance.
- Homework for Friday: Read the rest of
section 8.3, pages 271-274. Do the first problem on the
complex
analysis qualifying examination from January 2004: namely, show
that the images under stereographic projection of two non-zero
complex numbers z and w are diametrically
opposite points of the sphere if and only if the product of
z and the conjugate of w equals -1.
- Friday, January 30
- We worked on an exercise about unifying the
Weierstrass and Mittag-Leffler theorems.
- Homework for Monday: Do exercise 20 on
page 277 (Chapter 8), write up part 3 of the above
handout, and read the beginning of
section 9.1, pages 279-281.
- Monday, February 2
- We worked on an exercise on Jensen's
formula.
- Homework for Wednesday: Read the remainder of
section 9.1. Each group will prepare a derivation of the
Poisson-Jensen formula (exercises 1 and 2 on page 296 of
Chapter 9).
- Wednesday, February 4
- We discussed the Poisson-Jensen formula and its application in
the theory of entire functions.
- Homework for Friday: Read the first part of
section 9.3, pages 288-290. Do exercise 8 on
page 296 (Chapter 9) and problem 9 on the complex
analysis qualifying examination from January 2004.
- Friday, February 6
- We discussed the notion of order for entire functions and the
Hadamard factorization theorem for entire functions of finite
order.
- Homework for Monday: Finish reading section 9.3
and do exercises 3 and 10 on pages 296-297
(Chapter 9).
- Monday, February 9
- We discussed Hadamard's factorization theorem for entire
functions: the genus μ and the order λ are
related by the double inequality
μ ≤ λ ≤ μ+1.
- Homework for Wednesday: Complete the proof of
Hadamard's factorization theorem by proving the second half of the
inequality (which is exercise 12 on page 297).
- Wednesday, February 11
- We looked at the proof that the order of a canonical product
equals the convergence exponent of the zeroes; a version of the
Weierstrass and Mittag-Leffler theorems with essential
singularities; and Runge's approximation theorem.
- Homework for Friday: Read section 12.1, pages 361-367,
and do exercise 6 on page 380 (Chapter 12).
- Friday, February 13
- We worked on an exercise on Runge's
theorem.
- Homework: One group will prepare a presentation for
Monday on the Hadamard gap theorem (section 9.2), and the
other group will prepare a presentation for Wednesday on
Mergelyan's theorem (section 12.2).
- Monday, February 16
- We discussed the statement and the proof of Hadamard's gap
theorem.
- Homework for Wednesday: The first group will find an
example of a gap series that has the unit circle as natural
boundary and whose derivatives of all orders extend continuously to
the closed unit disc; the second group will complete the
preparation of a presentation on Mergelyan's theorem.
- Wednesday, February 18
- We discussed the proof of Mergelyan's theorem on polynomial
approximation.
- Homework for Friday: Read section 12.3, pages
376-379, and do exercise 11 on page 380
(Chapter 12).
- Friday, February 20
- We worked on an exercise on Swiss
cheese.
- Homework for Monday: (1) show that the analytic
capacity of a line segment is one-quarter of its length;
(2) show that the analytic capacity of the arc of the unit
circle from angle -φ to angle φ equals
sin(φ/2) when 0<φ<π.
- Monday, February 23
- We worked on an exercise on univalent
functions.
- Homework for Wednesday: Read section 13.1, pages
383-390, and solve the following exercise. Extract from the proof
on page 385 that the map z+z-1 maps the region {z:
|z|>r>1} onto the exterior of an ellipse (depending
on r). Deduce that the analytic capacity of the ellipse
(x/a)2+(y/b)2=1 is (a+b)/2.
- Wednesday, February 25
- We discussed proofs of the area theorem and of the estimate for
the second coefficient of a normalized schlicht function in the
unit disc.
- Homework for Friday: Do exercises 9 and 10 on
page 411 (Chapter 13).
- Friday, February 27
- We showed several ways that the class of normalized schlicht
functions in the unit disc is a normal family: by using the Koebe
one-quarter theorem, by using the coefficient estimates from the
Bieberbach conjecture (the de Branges theorem), and by using the
sharp growth estimate |z|/(1+|z|)2 ≤ |f(z)| ≤
|z|/(1-|z|)2.
- Homework for Monday: Read sections 10.1 and 10.2,
pages 299-307.
- Monday, March 1
- We discussed methods for analytic continuation.
- Homework for Wednesday: Read sections 10.3 and 10.4,
pages 307-314.
- Wednesday, March 3
- We discussed the monodromy theorem and the idea behind the
construction of the modular function using the reflection
principle.
- Homework for Friday: one group will prepare a
presentation on the modular function (Theorem 10.5.4), and the
other group will prepare a presentation on the proof of Picard's
little theorem via the modular function and the monodromy theorem
(Theorem 10.5.5).
- Friday, March 5
- We discussed the congruence subgroup of the modular group, its
fundamental domain, and the construction of the modular function
that is invariant under the action of the congruence subgroup.
- Homework for Monday: Work the exercise on the modular group handed out in
class.
- Monday, March 8
- We discussed the proof of Picard's little theorem via the
modular function and the monodromy theorem.
- Homework for Wednesday: do the exercise on Picard's theorems.
- Wednesday, March 10
- We discussed homework problems about Picard's theorem and the
modular group.
- Homework for Friday: Read section 13.2, pages 390-398,
about the boundary behavior of conformal maps.
- Friday, March 12
- We discussed two examples that answer questions posed in class
last time: (1) a transcendental entire function whose modulus
tends to infinity along every ray starting at the origin;
(2) a transcendental entire function that tends to zero along
every ray starting at the origin.
- There is no homework assignment over spring break.
- Monday, March 22
- We discussed properties equivalent to simple connectivity in
the plane.
- Homework for Wednesday: Read sections 11.1
and 11.2, pages 335-342.
- Wednesday, March 24
- We worked on an exercise on simple
connectivity.
- Homework for Friday: Read sections 11.3 and 11.4,
pages 343-349, and do exercises 4 and 5 on page 352.
- Friday, March 26
- We discussed the homology version of Cauchy's integral formula
and the connection between homotopy and homology.
- Homework for Monday: Read section 11.5, pages
349-352, and do exercise 31 on page 358.
- Monday, March 29
- We discussed three definitions of the Γ function and
some properties of the function.
- Homework for Wednesday: Read section 15.1, pages
447-455.
- Wednesday, March 31
- We worked on the identity relating the Γ function to
the sine function and the duplication formula for the
Γ function (Exercises 2 and 3 on page 465 in
the textbook).
- Homework for Friday: Read the beginning of
section 15.2, pages 455-459, and do the following problem.
Prove that
Γ(z) - ∑n≥0 (-1)n/(n!(z+n))
=
∫1∞ tz-1 e
-t dt.
- Friday, April 2
- We discussed an integral representation for the
Γ function via integration over the Hankel contour.
- Homework for Monday: Read the rest of
section 15.2, pages 460-464. Prove by a residue calculation
that when k is an integer, the integral over the Hankel contour of
tz-1/(et-1) is: (a) zero when k is
an integer greater than 1; (b) 2πi when k is 1;
(c) non-zero when k is 0; (d) zero when k is a
negative even integer. (The integral is not zero when k is a
negative odd integer, but you do not need to prove that.)
- Monday, April 5
- We discussed an integral representation for the
ζ function, the functional equation for the
ζ function, the trivial zeroes of the
ζ function, and the values of the ζ function at
the positive even integers.
- Homework for Wednesday: Prove that the function
ξ(z) defined by z(z-1)π-z/2ζ(z)Γ(z/2)
is entire and satisfies the functional equation
ξ(1-z)=ξ(z).
- Wednesday, April 7
- We discussed properties of Riemann's ζ and ξ functions,
the Riemann hypothesis, and the prime number theorem.
- Homework for reading day, Friday April 9: read
section 10.6, pages 323-330, about elliptic functions.
- Monday, April 12
- We discussed the construction of the Weierstrass
℘ function.
- Homework for Wednesday: Prove the duplication formula
for the Weierstrass ℘ function:
℘(2z)=(1/4)(℘″(z)/℘′(z))2−2℘(z).
- Wednesday, April 14
- We discussed how the modular group and the congruence subgroup
appear in the theory of the Weierstrass
℘ function.
- Homework for Friday: Prove that D=0, where D is
the determinant of the matrix whose first row is (℘(z),
℘′(z),1), whose second row is (℘(w),
℘′(w),1), and whose third row is (℘(z+w),
−℘′(z+w),1).
- Friday, April 16
- We worked on an exercise on the Weierstrass
℘ function.
- Homework for Monday: Finish that exercise (not to hand
in). The two groups should start looking at the exercises on Hadamard's three-circles theorem and on
Phragmén-Lindelöf theory to be presented in class
next Friday.
- Monday, April 19
- We discussed how the mapping (℘,℘′) gives a
bijection between a torus (the complex plane modulo a lattice) and
an elliptic curve and how the group law on the torus induces a
group law on the elliptic curve.
- Homework: for Wednesday and Friday, prepare the
presentations to be given in class Friday; for next Monday, read
the first part of Chapter 16, pages 469-474.
- Wednesday, April 21
- We discussed the question of what properties of a holomorphic
function defined by a power series can be recovered from the
moduli of the coefficients of the series.
- Friday, April 23
- The groups presented the variations on the maximum principle
(the three-circles theorem and Phragmén-Lindelöf
theory).
- Homework for Monday: Read the first part of
Chapter 16, pages 469-474.
- Monday, April 26
- We discussed the statement of the prime number theorem and some
numerical evidence for it; the Skewes number; and the three
functions ζ, Φ, and ϑ that appear in the proof of
the prime number theorem.
- Homework for Wednesday: Read pages 475-480 in
Chapter 16.
- Wednesday, April 28
- We continued the discussion of the proof of the prime number
theorem and the way that a Tauberian theorem for the Laplace
transform arises in the proof.
- Homework for Friday: Finish reading
Chapter 16.
- Friday, April 30
- We discussed Abelian and Tauberian theorems for power series
and analogous theorems for Laplace transforms.
- Homework for Monday: Prepare a list of statements of
theorems from this semester that have appeared on past qualifying
examinations. The final examination will be to state and prove a
subset of those theorems.
- Monday, May 3
- We discussed the major theorems of the semester (to appear on
the final examination).
- Tuesday, May 4
- This redefined day was our last class meeting. We proved
Montel's theorem (using the modular function), thus completing the
proof of Picard's big theorem. The lecture
notes are available.