Argument Principle and Rouche's Theorem

State and prove Rouché’s theorem.

State and prove Rouché’s theorem.

Suppose that \(f\) is holomorphic on the upper half plane \(\mathbb {H}=\{\operatorname {Im}(z)>0\}\), \(f\) extends continuously to \(\overline {\mathbb {H}}=\mathbb {H}\cup \mathbb {R}\), and \(f(\mathbb {R})\subset \mathbb {R}\). Show that \(f\) has a unique holomorphic extension to \(\mathbb {C}\) and explicitly identify that extension (in terms of \(f\)).

How many zeros does the function

have in the right half plane \(\{\operatorname {Re}(z)>0\}\).

State and prove Rouché’s theorem.

Let \(f(z)=1+3z+z^n\).

(a) Show that \(f\) has exactly one root in the unit disk \(\{z:|z|<1\}\).

(b) Show that all roots of \(f\) lie in the disk \[ \left \{z:|z|\le \frac {3+\sqrt {13}}{2}\right \}. \]

Using Rouché’s theorem find the number of zeros of the polynomial \(2z^5-z^3+3z^2-z+8\) in the region \(\{z\mid |z|>1\}\).

Prove Schwarz’s lemma: If \(f:\Delta \to \Delta \) is a holomorphic function from the open unit disk \(\Delta \) to itself with \(f(0)=0\) show that \(|f(z)|\le |z|\), \(|f'(0)|\le 1\) and if \(|f(z)|=|z|\) for some \(z\ne 0\) or \(|f'(0)|=1\) then \(f(z)=\lambda z\) with \(|\lambda |=1\).

(Hint: Apply the maximum principle to the function \(f(z)/z\).)

Let \(\alpha >1\) be arbitrary. Show that the equation \[ \alpha -z-e^{-z}=0 \] has exactly one solution in the half plane \(\{z:\operatorname {Re}z>0\}\), and moreover, this solution is real.

Let \(f\) be a function holomorphic on \(D\) and continuous on \(\overline {D}\). Assume that \(|f(z)|=1\) whenever \(|z|=1\). Show that \(f\) can be extended to a meromorphic function on the whole \(\mathbb C\), with at most finitely many poles.

(Hint: Starting with the Schwarz reflection principle for the upper half plane, deduce that an appropriate reflection continuation that can be used in this setting is \(z\mapsto \frac {1}{f(1/\bar z)}\).)

Let \(H=\{z\in \mathbb C:\operatorname {Im}z>0\}\) be the upper half plane and let \(\Delta =\{z\in \mathbb C:|z|<1\}\) be the unit disk.

a. Find a conformal map from \(H\) to \(\Delta \) that takes \(i\) to \(0\).

b. Let \(f:H\to H\) be a holomorphic map with \(f(i)=i\). Show that \(|f'(i)|\le 1\) and that if \(|f'(i)|=1\) then \(f\) is of the form \(f(z)=\frac {az+b}{cz+d}\) with \(a,b,c,d\in \mathbb R\).

Let \(U\) be the open unit disc in \(\mathbb C\). Let \(U^+\) be the intersection of \(U\) with the upper half plane.

(a) Let \(f(z)=\frac {1}{2}\left (z+\frac {1}{z}\right )\). Describe \(f(U^+)\).

(b) Give a conformal equivalence of \(U^+\) and \(U\).

Let \(\lambda >1\). Use Rouche’s Theorem to show that the equation \(\lambda -z-e^{-z}=0\) has one solution for \(\operatorname {Re}(z)>0\).