MATH 406: Problem Set 12

due Friday, April 22, 2005

(Revised version)

1. Find the roots of the following polynomials; show your work, recommended not to use a calculator!

a)    b)    c)

2. Fisher 4.2.1

3. Fisher 4.2.2

4. Find the electrostatic potential between two infinite flat plates in the plane: one along the positive real axis, at , and one starting at the origin and situated up and to the left, meeting the first plate at 120 , held at . Note that there is a small insulator between the two plates at the origin!

5. Consider two non-concentric circles, one of which crosses the axis at and (with its center on the real axis), and one defined by . First show that the following map

a) moves the inner circle to , and b) keeps the outer circle fixed ( ). Next find the electrostatic potential between the two original cylinders in the -plane, with the inner one held fixed at V and the outer one held fixed at V.

6. (F 4.2.15) Using the Milne-Thompson Circle Theorem and the Zhukovsky map, obtain the complex function for flow past an ellipse with axes and (see Fisher p.282).

Version 1.1