Impedance

The subject of Raoul Bott's first paper [1] dates back to his engineering days. An electrical network determines an impedance function $Z(s)$ , which describes the frequency response of the network. This impedance function $Z(s)$ is a rational function of a complex variable $s$ and is positive-real (p.r.) in the sense that it maps the right half-plane into itself. An old question in electrical engineering asks whether conversely, given a positive-real rational function $Z(s)$ , it is possible to build a network with $Z(s)$ as its impedance function. In some sense O. Brune had solved this problem in 1931, but Brune's solution assumes the existence of an ``ideal transformer,'' which in practice would have to be the size of, say, the Harvard Science Center. The assumption of an ideal transformer renders Brune's algorithm not so practical, and it was Raoul's dream at McGill to remove the ideal transformer from the solution.

At his first meeting with his advisor Richard Duffin at Carnegie Tech, he blurted out the problem right away. Many days later, after a particularly fruitless and strenuous discussion, Raoul went home and realized how to do it. He called Duffin. The phone was busy. As it turned out, Duffin was calling him with exactly the same idea! They wrote up the solution to the long-standing problem in a joint paper, which amazingly took up only two pages.