Using

Rouche's theorem, it can be easily shown that [F.sup.m](x) has a unique zero [[alpha].sub.m] satisfying [absolute value of [[alpha].sub.m]] [less than or equal to] 2 (See Table III for the approximate values of [[alpha].sub.m] for odd m [less than or equal to] 7).

Among their topics are the generalized argument principle and

Rouche's theorem, perturbations of cavities and resonators, photonic and phononic band gaps, plasmonic resonances for nanoparticles, near-cloaking, and Helmholtz resonators, and Minnaert resonances for bubbles.

More importantly, we drop the lower order term in (35) and then consider

Rouche's Theorem for the identity

Rouche's theorem shows that R(z, [z.sup.h]) has exactly one simple root in the disk {z : |z| [less than or equal to] [zeta] + [kappa]} for sufficiently large h.

Rouche's theorem shows that R(z, [z.sup.h]) has exactly one simple root in the disk {z : [absolute value of z] [less than or equal to] [zeta] + [kappa]} for sufficiently large h.

Then, by

Rouche's Theorem, [PHI]([lambda]) and [PHI]([lambda]) - [PHI]([lambda]) have the same number of zeroes in an open unit disk [absolute value of A] < 1.

Therefore, by using

Rouche's theorem [3], there is a unique positive [[tau].sub.k] = [[tau]'.sub.k] + o(1/k) satisfying (26), that is, the characteristic equation (10) has a pair of purely imaginary roots of the form [+ or -]i[[omega].sub.0] as k [right arrow] [infinity].

By

Rouche's theorem we obtain 1/[??]([xi]) has a zero only at [??] = 0 of multiplicity l.

By

Rouche's Theorem, the transcendental equation (i) has roots with positive real parts if and only if it has purely imaginary roots.

One involves control over derivatives and/or complex analyticity which will allow uniqueness via an appeal to an intermediate value theorem or a use of

Rouche's theorem. These will each require extra hypotheses on the Verblunsky coefficients or Jacobi parameters.

Because [h.sub.2](t) has the unique zero t = 1/2 in the disk [absolute value of t] < r, it follows from

Rouche's theorem that h(t) also has the unique zero in the disk [absolute value of t] < r.Since this holds for any r < 1 sufficiently close to 1, it means that h(t),hence g(t) has the unique zero in the unit disk [absolute value of t] < 1.

Klimenok, On the modification of

Rouche's Theorem for the Queueing theory problems, Queueing Systems 38(2001), 431-434.