Read sections 1.1, 1.2, and 1.3, pages 1-14.
Do exercises 6, 9, 23, and 29, pages 22-25 (Chapter 1).
Read sections 1.4 and 1.5, pages 15-21.
Do exercises 30, 32, 37, and 47, pages 25-27 (Chapter 1).
Read section 2.1 and the first part of section 2.2, pages 30-39.
Do exercises 10, 18, and 52, pages 23, 24, and 28 (Chapter 1).
Do exercise 1, page 62 (Chapter 2).
Read the end of section 2.2 and section 2.3, pages 40-45.
Read section 2.4, pages 45-53.
Do exercises 36 and 49, pages 26 and 28 (Chapter 1).
Do exercises 3, 5, and 9, pages 63-64 (Chapter 2).
Extend Proposition 2.1.6, page 33, to the case of a continuously differentiable function f that is not necessarily holomorphic.
Read sections 2.5 and 2.6, pages 53-62.
Read section 3.1, pages 71-76.
Do exercises 15, 17, 23, and 25 on pages 64-67 (Chapter 2).
Read section 3.2, pages 76-83.
Do exercises 16, 22, and 30 on pages 65-68 (Chapter 2).
Prove that for every positive integer n, the integral over the circle |z|=n+1 of the reciprocal of (z-1)2(z-2)2···(z-n)2 is zero.
Read sections 3.3 and 3.4, pages 83-90.
Do exercises 19 and 44 on pages 24 and 27 (Chapter 1).
Do exercises 7 and 24 on pages 97 and 99 (Chapter 3).
Read section 3.5, pages 90-93.
Do exercises 11e, 11f, 11g, and 19 on page 98 (Chapter 3).
Show that the sum over n of zn/n, which is a series with radius of convergence equal to 1, diverges when z=1 and converges for every other point on the circle where |z|=1.
Hint: use Dirichlet's convergence test. This says that if {an} is a sequence (of complex numbers) whose partial sums are bounded, and if {bn} is a monotonically decreasing sequence of real numbers with limit 0, then the infinite series with general term anbn converges.
Read section 3.6, pages 93-96.
Do exercises 9, 13, 17, and 20(a) on pages 97-99 (Chapter 3).
Do exercises 15, 16, 20(d), and 27, pages 98-100 (Chapter 3).
Make a list of the five most important theorems in Chapters 1-3. For each of the five theorems, write one sentence stating the key idea in the proof of the theorem.
Read section 4.1, pages 106-110.
Do exercises 26, 29, 37, and 44, pages 99-102 (Chapter 3). Notice that in problem 44, the function f is not known a priori to be continuous.
Read section 4.2, pages 110-114.
Do exercises 39 and 42, page 102 (Chapter 3).
Do exercises 3 and 4, page 147 (Chapter 4).
Read section 4.3, pages 114-120.
Read sections 4.4 and 4.5, pages 120-129.
Do exercises 17b, 29d, 35a, and 36a on pages 150-152 (Chapter 4).
Read section 4.6, pages 130-140.
Do exercises 14, 35c, 36e, and 42, pages 149-154 (Chapter 4).
Do exercises 49, 50, and 52, page 156 (Chapter 4) and check your answers with Maple or Mathematica.
Read section 4.7, pages 140-146.
Do exercises 61 and 68, page 157 (Chapter 4).
Read section 5.1, pages 159-164.
Do exercises 9, page 148, and 57, page 156 (Chapter 4).
Read section 5.2, pages 164-168.
Use contour integration to prove that the sum of the squares of the reciprocals of the positive fixed points of the tangent function is 1/10.
Read section 5.3, pages 168-171.
Finish the problem from last time.
Read section 5.4, pages 172-173.
Do exercises 5, 10a, and 16, pages 177-180 (Chapter 5).
Read section 5.5, pages 173-176.
Read sections 6.1 and 6.2, pages 181-186.
Solve the following problems.
Determine which biholomorphic functions from the unit disk onto itself have a fixed point in the interior of the disk.
Prove that if a holomorphic (not necessarily biholomorphic) function from the unit disk into (not necessarily onto) the unit disk has two fixed points in the interior of the disk, then it is the identity function.
Read section 6.3, pages 186-191.
Solve the following related problems.
Show that there is no holomorphic function whose domain is the whole complex plane and whose range is the whole unit disk.
Show that there is a holomorphic function whose domain is the whole unit disk and whose range is the whole complex plane.
Do exercise 21, page 206 (Chapter 6).
Read the rest of Chapter 6.
Read sections 7.1 and 7.2, pages 208-213.
Read sections 7.3 and 7.4, pages 213-222.