WJD Open Notebook

2003 November 17

Conventions. Region means open connected set and holomorphic means analytic.

1. (a) State the Cauchy-Riemann equations.
   (b) Prove the following using those equations.
   Let $f \colon \Omega \to \mathbb C$ be a holomorphic function defined on an region $\Omega \subset \mathbb C$ and let $h \colon \mathbb R^2 \to \mathbb R$ be a smooth function. Let $L = \{(x,y) : h(x, y) =\lambda \}$, where $\lambda \in \mathbb R$ and suppose $\nabla h \neq 0$ anywhere in $L$. If $f(z) \in L$ for all $z\in \Omega$, then $f$ is a constant function on $\Omega$. (Of course, $\mathbb C = \mathbb R^2$ in the usual fashion.)

2. Suppose that $f$ is a holomorphic function defined in the open unit disk $D$ such that $\mathrm{Re}\ f(z) > 0$, for all $z\in D$, and $f(0) = 1$. Show that for all $z\in D$, $$\frac{1-|z|}{1+|z|} \leq |f(z)| \leq \frac{1+|z|}{1-|z|}.$$ It is good enough to establish either inequality since both are obtained in the same fashion. (Hint: Apply Schwarz's Lemma.)

3. Suppose $f$ is holomorphic in the sector $\Sigma = \{z \in \mathbb C : \alpha_1 < \mathrm{arg}\ z < \alpha_2,\ \alpha_2-\alpha_1 < 2\pi,\ 0< |z|<\infty \}$. Suppose there exist positive real numbers $\delta$, $\epsilon$, $C_1$, $C_2$ such that, for all $z \in \Sigma$, $$|f(z)| < \left\{\begin{array}{ll} C_1|z|^{-1+\delta}, & 0 < |z| < 1,\\ C_2|z|^{-1-\epsilon}, & 1 < |z|.\end{array}\right.$$ (a) The integral $I_\alpha =\int_0^\infty f(z)\ dz$ converges along the ray $\arg z = \alpha$, $\alpha_1 < \alpha < \alpha_2$; prove its value is independent of $\alpha$.
   (b) Suppose that $f(z) = \frac{1}{z_3+1}$ so that $$I_\alpha =\int_0^\infty \frac{1}{z_3+1}\ dz, \quad (\arg z = \alpha).$$ Clearly $I_\alpha$ is well defined unless $\alpha$ is an odd multiple of $\pi/3$. Show that $I_\alpha$ has three distinct values and find $I_0 - I_{2\pi/3}$. The result need not be simplified.
   (c) Determine the value of $I_0 = \int_0^\infty \frac{1}{r_3+1}\ dr$. The result need not be simplified. (Hint: Use Part (b) and directly compare the line integrals $I_0$ and $I_{2\pi/3}$.

4. Let $\Omega$ be a region in $\mathbb C$ and $z_0\in \Omega$. Let $D$ be the open unit disk.
   (a) Under what conditions on $\Omega$ does there exist a one-to-one holomorphic function $f$ that maps $\Omega$ onto $D$? State the definitions of any topological conditions imposed on $\Omega$ in order for $f$ to exist.
   (b) Assuming $\Omega$ satisfies the necessary conditions for $f$ to exist, if we require that $f(z_0) = 0$ and $f'(z_0)> 0$ then it is known that the holomorphic function $f$ is uniquely determined. Suppose moreover that $\Omega$ is symmetric with respect to the real axis, i.e., $\bar{\Omega} = \Omega$, and $z_0$ is real. Show that the uniquely determined function $f$ satisfies the following: $f(\bar{z}) = \overline{f(z)}$, for all $z \in \Omega$, and $f^{(n)}(z_0)$ is real for all $n \geq 0$.