Math 617 -- Homework Assignments

• Assignment 1. Week of 9/1 -- due Wednesday, 9/10.
  1. Section 1.2 (page 3): 3, 4, 5
  2. Section 1.4 (page 6): 2(b), 3
  3. Section 1.5 (page 7): 1
  4. Section 1.6 (page 10): 4
  5. Section 2.1 (page 13): 6, 11

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• Assignment 2. Week of 9/8 -- due Wednesday, 9/17.
  1. Section 2.4 (page 24): 2, 4
  2. Section 2.5 (page 28): 2, 3
  3. Section 2.6 (page 29): 1
  4. Section 3.1 (page 33): 1, 6(b,d)
  5. Section 3.2 (page 43): 1, 4

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• Assignment 3. Week of 9/15 -- due Friday, 9/26.
  1. Section 3.2 (page 44): 7, 8, 10, 11, 14, 18-20
  2. Use the Cauchy-Riemann conditions to verify that exp(z):=exp(x)*(cos(y) + i*sin(y)) is anaytic everywhere.
  3. Derive the CR conditions in polar coordinates. Use them to directly show that the principal branch of log(z) is analytic.

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• Assignment 4. Week of 9/22 -- due Friday, 10/3.
  1. Section 3.3 (page 54): 1, 4, 5, 6, 8, 9, 13, 14, 17, 18

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• Assignment 5. Week of 10/13 -- due Friday, 10/24.
  1. Section 4.1 (page 67): 1, 7, 9, 11, 13, 16, 18(c), 19, 21, 22
  2. 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.)

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• Assignment 6. Week of 10/20 -- due Friday, 10/31.
  1. Section 4.2 (pages 73-75): 5, 6, 7, 9(b,e), 10, 11
  2. 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.
  3. 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.)

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• Assignment 7. Week of 10/27 -- due Friday, 11/7.
  1. Section 4.3 (page 80): 1, 3, 5, 6, 9
  2. Section 4.4 (page 83): 3, 4
  3. Section 4.5 (page 87): 3-7

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• Assignment 8. Week of 11/17 -- due Wednesday, 11/26.
  1. Section 4.7 (page 100): 4
  2. Section 5.1 (page 110): 1(d,e,h), 4, 5, 7, 12(a), 13
  3. Section 5.2 (page 121): 1(b,c), 2(c,d)

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• Assignment 9. Week of 11/24 -- due Wednesday, 12/3.
  1. Section 5.2 (page 121): 1(d), 2(e,g), 4, 5, 6, 12
  2. Section 5.3 (page 126): 1, 2, 3

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• Assignment 10. Week of 12/1 -- due Wednesday, 12/10.
  1. Section 6.1 (page 129): 2, 4, 7, 8
  2. Section 6.2 (page 132): 3, 5, 6
  3. Section 6.3 (page 137): 5, 6, 7