Math 618 -- Homework Assignments
Assignment 1. Week of 1/21.
- Read Section 10.1 and review integration theory.
Assignment 2. Week of 1/26 -- due Wednesday, 2/4.
- Section 10.1 (page 255): 1, 5, 7, 8, 11
- Section 10.2 (page 262): 1, 3(a,b,c)
Assignment 3. Week of 2/9 -- due Friday, 2/20.
- 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.
- 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+.
- 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.
- 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.
- 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.
- Section 6.3 (page 137): 3
- Section 7.5 (page 173): 1, 4, 10
- Section 7.6 (page 176): 2, 4
Assignment 5. Due Wednesday, 4/15.
- Section 7.7 (page 185): 2, 3
- Section 7.8 (page 194): 1, 2, 4
Assignment 6. Due Wednesday, 5/1.
- Section 8.1 (page 201): 2
- Section 8.3 (page 206): 2(a,e,f), 5, 6
- Section 9.1 (page 213): 1, 2
Updated: April 22, 1998