Examination 6¶
2004 Apr¶
Instructions. Use a separate sheet of paper for each new problem. Do as many problems as you can. Complete solutions to five problems will be considered as an excellent performance. Be advised that a few complete and well written solutions will count more than several partial solutions.
Notation. \(D(z_0,R) = \{z\in \mathbb{C}: |z-z_0|<R\}\) \(R>0\). For an open set \(G\subseteq \mathbb{C}\), \(H(G)\) will denote the set of functions which are analytic in \(G\).
Let \(\gamma\) be a rectifiable curve and let \(\varphi \in C(\gamma^*)\). (That is, \(\varphi\) is a continuous complex function defined on the trace, \(\gamma^*\), of \(\gamma\).)
Let \(F(z) = \int_\gamma \frac{\varphi(\omega)}{(\omega-z)} \, d\omega, \quad z\in \mathbb{C}\setminus \gamma^*\).
Prove that \(F'(z) = \int_\gamma \frac{\varphi(\omega)}{(\omega-z)^2} \, dw, \quad z\in \mathbb{C}\setminus \gamma^*\), without using Leibniz’s Rule.
State the Casorati-Weierstrass theorem.
Evaluate the integral
\[I =\frac{1}{2\pi i} \int_{|z|=R} (z-3) \sin\left(\frac{1}{z+2}\right)\, dz \; \text{ where } R\geq 4.\]
Let \(f(z)\) be an entire function such that \(f(0)=1\), \(f'(0)=0\) and
Prove that \(f(z)=1\) for all \(z\in \mathbb{C}\).
Let \(C\) be an arbitrary circle through \(-1\) and \(1\). Suppose that \(z_1\) and \(z_2\) are two points which do lie on the circle \(C\) and satisfy \(z_1z_2 = 1\). Show that one of these points lies inside \(C\) and the other lies outside \(C\).
Show that there is no one-to-one analytic function that maps \(G = \{z : 0 < |z| < 1\}\) onto the annulus \(\Omega = \{z : r < |z| < R\}\), where \(r>0\).
State a theorem that gives a sufficient condition for a family \(\mathcal F\) of analytic functions to be normal in a domain \(G\).
Let \(\mathcal F\subseteq H(D)\) be a family of analytic functions on the open unit disk \(D = D(0,1)\). Let \(\{M_n\}\) be a sequence of positive real numbers such that \(\varlimsup_{n\to \infty} \sqrt[n]{M_n} < 1\). If for each \(f(z) = \sum_{n=0}^\infty a_n z^n \in \mathcal F\), \(|a_n| \leq M_n\) for all \(n\), prove that \(\mathcal F\) is a normal family.
Is there a harmonic function \(u(z)\) defined on the open unit disk, \(D(0, 1)\), such that \(u(z_n) \to \infty\) whenever \(|z_n|\to 1^-\)? Prove your answer.
Let \(G\) be a simply connected domain with at least 2 boundary points. Let
Prove, without using the Riemann mapping theorem, that the set \(S\) is nonempty.
Solutions¶
Solution to Problem 34
Solution to Problem 35
Solution to Problem 36
Solution to Problem 37. (coming soon)
Solution to Problem 38.
Solution to Problem 39.
Solution to Problem 40. (coming soon)
Solution to Problem 41. (coming soon)
Footnotes
- 1
I was fortunate to have worked on this exam after having just read a beautiful treatment of the Hadamard factorization theorem in Stein and Sharkachi’s book [SS03]. If you need convincing that this theorem is worth studying, take a look at how easily it dispenses with this and other, otherwise challenging exam problems. Stein and Sharkachi seem to have set things up just right, so that the theorem is very easy to apply.
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