Math 617 -- Homework Assignments
- Assignment 1. Week of 9/1 -- due Wednesday, 9/10.
- Section 1.2 (page 3): 3, 4, 5
- Section 1.4 (page 6): 2(b), 3
- Section 1.5 (page 7): 1
- Section 1.6 (page 10): 4
- Section 2.1 (page 13): 6, 11
- Assignment 2. Week of 9/8 -- due Wednesday, 9/17.
- Section 2.4 (page 24): 2, 4
- Section 2.5 (page 28): 2, 3
- Section 2.6 (page 29): 1
- Section 3.1 (page 33): 1, 6(b,d)
- Section 3.2 (page 43): 1, 4
- Assignment 3. Week of 9/15 -- due Friday, 9/26.
- Section 3.2 (page 44): 7, 8, 10, 11, 14, 18-20
- Use the Cauchy-Riemann conditions to verify that
exp(z):=exp(x)*(cos(y) + i*sin(y)) is anaytic everywhere.
- Derive the CR conditions in polar coordinates. Use them to
directly show that the principal branch of log(z) is analytic.
- Assignment 4. Week of 9/22 -- due Friday, 10/3.
- Section 3.3 (page 54): 1, 4, 5, 6, 8, 9, 13, 14, 17, 18
- Assignment 5. Week of 10/13 -- due Friday, 10/24.
- Section 4.1 (page 67): 1, 7, 9, 11, 13, 16, 18(c),
19, 21, 22
- Show that if f is analytic in a region G that contains a triangle
T defined by z1, z2, and z3, and if T is traversed once in the
counterclockwise direction, then the intergal of f over T is 0. (Hint:
By the way we have defined ``analytic'', you may use Green's Theorem
in the plane here.)
- Assignment 6. Week of 10/20 -- due Friday, 10/31.
- Section 4.2 (pages 73-75): 5, 6, 7, 9(b,e), 10, 11
- Find the Taylor series for log(z) about z=-1+i. What is the
radius of convergence of this series? Evaluate this series at
z=-1-i/4.
- Show that the Cauchy-Goursat Theorem still holds if we assume
that $f$ is continuous, rather than differentiable, at a finite number
of points in or on T. (Hints: (1) It suffices to do one point. Why?
(2) By selecting the appropriate triangles, one may make use of the
standard version of the C-G Theorem.)
- Assignment 7. Week of 10/27 -- due Friday, 11/7.
- Section 4.3 (page 80): 1, 3, 5, 6, 9
- Section 4.4 (page 83): 3, 4
- Section 4.5 (page 87): 3-7
- Assignment 8. Week of 11/17 -- due Wednesday, 11/26.
- Section 4.7 (page 100): 4
- Section 5.1 (page 110): 1(d,e,h), 4, 5, 7, 12(a), 13
- Section 5.2 (page 121): 1(b,c), 2(c,d)
- Assignment 9. Week of 11/24 -- due Wednesday, 12/3.
- Section 5.2 (page 121): 1(d), 2(e,g), 4, 5, 6, 12
- Section 5.3 (page 126): 1, 2, 3
- Assignment 10. Week of 12/1 -- due Wednesday, 12/10.
- Section 6.1 (page 129): 2, 4, 7, 8
- Section 6.2 (page 132): 3, 5, 6
- Section 6.3 (page 137): 5, 6, 7
Updated: December 4, 1997