next up previous
Next: About this document ...

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

    $\displaystyle 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




next up previous
Next: About this document ...
Andrew L. Belmonte 2005-04-18