The Left-Half Plane

In the left-half plane, there are many potential poles:

• The first of the two equations
  $$e^\tau ( 1 - \tau) = \cos(\nu)$$
  has infinitely many solutions for each $\tau<0$, since the elementary inequality $1-\tau \leq e^{-\tau}$ implies
  $$e^\tau ( 1 - \tau) < e^\tau e^{-\tau} = 1$$

• The second equation,
  $$e^\tau = \frac{\sin(\nu)}{\nu}$$
  has an increasing number of solutions as $\tau$ grows more and more negative.

• As $\tau\to-\infty$, the number of solutions becomes infinite
  and are given by the zeros of $\sin(\nu)$

• At $\tau\to-\infty$, the solutions of the other equation converge to the zeros of $\cos(\nu)$

• Thus, the solutions of
  $$\begin{eqnarray*} e^\tau ( 1 - \tau) &=& \cos(\nu) \\ e^\tau &=& \frac{\sin(\nu)}{\nu} \end{eqnarray*}$$
  

  may not necessarily occur together for $\tau<0$, as they must.