Read section 8.1, pages 255-263.
Do exercises 6 and 13, page 277 (Chapter 8).
Read section 8.2, pages 263-267.
Do exercise 15 on page 278 (Chapter 8). You will want to refer to Example 4.6.6 on page 139.
Recall that in class, we deduced an infinite product representation for the sine function, but the details of the exponential factor were left open. This exercise completes the derivation.
Read the first part of section 8.3, pages 267-271.
Do exercise 21 on page 279 (Chapter 8).
Finish reading Chapter 8.
Do exercise 20, page 278 (Chapter 8).
Use the method discussed in class to give a new proof of Theorem 8.3.8 on page 275.
Read in section 9.1 from Theorem 9.1.5 on page 283 to page 286.
Read pages 280-283 in section 9.1.
Extend Jensen's formula to cover the case that f(0)=0.
The ideal subgroup will report on generalizing problem 20, page 278 (Chapter 8) to the case in which the entire functions do have common zeroes.
Do the exercise on Jensen's formula handed out in class.
Read in section 9.3, pages 289-292 (top) and 296-297.
Write a solution to part 4 of the class exercise.
Work the exercise on order and series coefficients distributed in class.
Read sections 10.1 and 10.2, pages 300-308.
Read sections 10.3 and 10.4, pages 308-315.
Do exercise 11, page 331 (Chapter 10).
Do exercise 34, page 151 (Chapter 4) about the spherical metric.
Use contour integration to show that the integral from 0 to infinity of x-a/(1+x) equals pi/sin(a pi) when 0<a<1. (Suggestion: keyhole contour.) Deduce that the integral in exercise 34 from last time continues analytically to a meromorphic function on the whole plane with poles exactly at the points 4k-1, where k is an integer.
Read pages 315-321 concerning the modular group and the modular function (Chapter 10).
Read pages 322-323 (proof of Picard's theorem). Question: where does the proof break down if you replace lambda with the exponential function and C \ {0,1} with C \ {0}?
Read the rest of Chapter 10 (pages 323-330, about elliptic functions).
Do exercises 7, 16, and 21 on pages 330-332 (Chapter 10).
Finish reading Chapter 11.
Complete part 3 of the handout from class.
Read the beginning of Chapter 12, pages 358-361.
Be prepared to present solutions to the exercise on the proof of Runge's theorem.
Read the conclusion of section 12.1, pages 361-364.
Be prepared to present solutions to the exercise on Alice Roth's Swiss cheese.
Be prepared to present solutions to the exercise on the d-bar equation.
The universal subgroup will report on the article by Charles K. Chui and Milton N. Parnes, Approximation by overconvergence of a power series, Journal of Mathematical Analysis and Applications 36 (1971) 693-696.
Do exercise 6 on page 378 (Chapter 12).
Do exercise 3 on page 377 (Chapter 12).
Read section 12.3, pages 374-376.
Compute the analytic capacity of a line segment in the plane. Does it make a difference whether it is the analytic capacity gamma or the continuous analytic capacity alpha?
Skim section 15.1 and make a list of as many different definitions of the Gamma function as you can.
Since the Gamma function has no zeroes, the reciprocal of the Gamma function is entire. What are its order and genus?
Do exercise 5 on page 478 (Chapter 15).
Read pages 457-459 in Chapter 15.
Do exercise 25 on page 482 (Chapter 15).
Complete the computation begun in class of the integral from 0 to infinity of tz-1 times cos(t) or sin(t). Be sure to pay attention to convergence issues.
Compute the residues of the Gamma function at its poles by using the analytic continuation of Gamma via contour integration, as developed in class.
Complete the derivation, begun in class, of the functional equation for the zeta function by proving that the integral over the square contour does tend to zero in the limit.
Use the functional equation for the zeta function to compute the values of the function at 0 and -1. You can check your answers with Maple, which has the zeta function built in.
Do exercise 2, page 477 (Chapter 15).
Make a list of ten significant theorems from this course.