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) $$\,z^2 + (2i-7)z -14i$$   b) $$\,z^2 + (i+8)z +8i$$   c) $$\,z^2 - (12i+9)z +108i \,$$

2. Fisher 4.2.1
   

3. Fisher 4.2.2
   

4. Find the electrostatic potential $\phi$ between two infinite flat plates in the plane: one along the positive real axis, at $\phi_0 = 0$ , and one starting at the origin and situated up and to the left, meeting the first plate at 120$^\circ$ , held at $\phi_1$ . 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 $x = 1/7$ and $x = 1/2$ (with its center on the real axis), and one defined by $\vert z_2\vert = 1$ . First show that the following map
   $$w = \frac{3z - 1}{3 - z}$$
   a) moves the inner circle to $\vert w_1\vert = 1/5$ , and b) keeps the outer circle fixed ($\vert w_2\vert = 1$ ). Next find the electrostatic potential $\phi$ between the two original cylinders in the $z$ -plane, with the inner one held fixed at $\phi_1 = 12$ V and the outer one held fixed at $\phi_2 = 40$ 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 $a$ and $b < a$ (see Fisher p.282).

Version 1.1