Math 136: Complex Variables and Integral Transforms

Homework 12

Read Sections 7.3 -- 7.6 of Saff and Snider,
and the summary at the end of Chapter 7.

Do these problems from Saff and Snider:

  • Sec. 7.1, Invariance of Laplace's Equation: 3.
    (Think about 1.)
  • Sec. 7.2, Geometric Considerations: 2, 10, 11c, e.
    For Question 2, the book suggests mimicking a proof which can now be found on p.120 in section 3.3 (typo in book says 3.2). The following fact is useful for using Theorem 3 to prove continuity: A function f:D->C is continuous on D if and only if whenever E is an open set in C, its inverse image f-1(E) is open in D. That is, iff `inverse images of open sets are open'.
    (Think about 7, 8, 9, and the rest of 11. The area formula in 9 uses the Jacobian from Section 2.4 Q15.)
  • Sec. 7.3, Moebius Transformations: 4, 11.
    (Think about 1 (see Example 2), 6, 7c.)

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