Math 618 -- Homework Assignments

Assignment 1. Week of 1/21.
  1. Read Section 10.1 and review integration theory.

Assignment 2. Week of 1/26 -- due Wednesday, 2/4.
  1. Section 10.1 (page 255): 1, 5, 7, 8, 11
  2. Section 10.2 (page 262): 1, 3(a,b,c)

Assignment 3. Week of 2/9 -- due Friday, 2/20.
  1. Find a function u that is harmonic in the unit disk and on the unit circle is 1 for theta = 0 to theta =¼*pi, and 0 for theta = ¼*pi to theta = 2*pi.
  2. Verify that w=sin(½pi*z) takes the strip in the upper half plane bounded by the lines x=-1, y=0, and x=1 conformally onto H+.
  3. Use the Schwarz-Christoffel transformation to map the H+ onto the region above the curve: z(t)=t+i, t <= 0; z(t)=(1-t)i, 0 < t < 1; z(t)=t-1, t >= 1.
  4. Complete the details needed to show that the Schwarz-Christoffel map that takes H+ into the square with corners {-1+i,-1,1,1+i} is actually univalent and onto.
  5. Let C be a closed piecewise smooth Jordan curve, and let let the interior of C be G. Let F:Dcl -> Gcl be analytic. In addition, let w(t)=F(eit) traverse C once counterclockwise as t goes from 0 to 2*pi. Show that F is a conformal map of D onto G.

Assignment 4. Due Monday, 4/6.
  1. Section 6.3 (page 137): 3
  2. Section 7.5 (page 173): 1, 4, 10
  3. Section 7.6 (page 176): 2, 4

Assignment 5. Due Wednesday, 4/15.
  1. Section 7.7 (page 185): 2, 3
  2. Section 7.8 (page 194): 1, 2, 4

Assignment 6. Due Wednesday, 5/1.
  1. Section 8.1 (page 201): 2
  2. Section 8.3 (page 206): 2(a,e,f), 5, 6
  3. Section 9.1 (page 213): 1, 2

Updated: April 22, 1998