Skip to content

2003 Nov

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 \(n \geq 0\).