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2001 Nov

2001 November 21

Instructions. Make a substantial effort on all parts of the following problems. If you cannot completely answer Part (a) of a problem, it is still possible to do Part (b). Partial credit is given for partial progress. Include as many details as time permits. Throughout the exam, \(z\) denotes a complex variable, and \(\mathbb C\) denotes the complex plane.

  1. (a) Suppose that \(f(z) = f(x+iy) = u(x,y) + i v(x,y)\) where \(u\) and \(v\) are \(C^1\) functions defined on a neighborhood of the closure of a bounded region \(G\subset \mathbb C\) with boundary which is parametrized by a properly oriented, piecewise \(C^1\) curve \(\gamma\). If \(u\) and \(v\) obey the Cauchy-Riemann equations, show that Cauchy's theorem \(\int_\gamma f(z) \ dz = 0\) follows from Green's theorem, namely

    \[\int_\gamma P\ dx + Q\ dy = \int_G \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\ dx\ dy,\]

    for \(C^1\) functions \(P\) and \(Q\).\ (b) Suppose that we do not assume that \(u\) and \(v\) are \(C^1\), but merely that \(u\) and \(v\) are continuous in \(G\) and

    \[f'(z_0) = \lim_{z\rightarrow z_0} \frac{f(z) - f(z_0)}{z-z_0}\]

    exists at some (possibly only one!) point \(z_0 \in G\). Show that given any \(\epsilon >0\), we can find a triangular region \(\Delta\) containing \(z_0\), such that if \(T\) is the boundary curve of \(\Delta\), then

    \[\left|\int_T f(z)\ dz\right| = \frac{\epsilon L^2}{2},\]

    where \(L\) is the length of the perimeter of \(\Delta\). (Hint: Note that part (a) yields \(\int_T (az+b) \ dz =0\) for \(a, b \in \mathbb C\), which you can use here in (b), even if you could not do Part (a). You may also use the fact that \(\left|\int_T g(z)\ dz\right| \leq L \cdot \sup\{|g(z)|:z\in T\}\) for \(g\) continuous on \(T\).)

  2. Give two quite different proofs of the Fundamental Theorem of Algebra that if a polynomial with complex coefficients has no complex zero, then it is constant. You may use independent, well-known theorems and principles such as Liouville's Theorem, the Argument Principle, the Maximum Principle, Rouche's Theorem, and/or the Open Mapping Theorem.

  3. (a) State and prove the Casorati-Weierstrass Theorem concerning the image of any punctured disk about a certain type of isolated singularity of an analytic function. You may use the fact that if a function \(g\) is analytic and bounded in the neighborhood of a point \(z_0\), then \(g\) has a removable singularity at \(z_0\).\ (b) Verify the Casorati-Weierstrass Theorem directly for a specific analytic function of your choice, with a suitable singularity.

  4. (a) Define \(\gamma : [0,2\pi] \rightarrow \mathbb C\) by \(\gamma(t) = \sin (2t) + 2i \sin (t)\). This is a parametrization of a "figure 8" curve, traced out in a regular fashion. Find a meromorphic function \(f\) such that \(\int_\gamma f(z) \ dz = 1\). Be careful with minus signs and factors of \(2\pi i\).\ (b) From the theory of Laurent expansions, it is known that there are constants \(a_n\) such that, for \(1<|z|<4\),

    \[\frac{1}{z^2 - 5z + 4} = \sum_{n=-\infty}^\infty a_n z^n.\]

    Find \(a_{-10}\) and \(a_{10}\) by the method of your choice.

  5. (a) Suppose that \(f\) is analytic on a region \(G\subset \mathbb C\) and \(\{z\in \mathbb C: |z-a|\leq R\} \subset G\). Show that if \(|f(z)| \leq M\) for all \(z\) with \(|z-a|=R\), then for any \(w_1, w_2\in \{z\in \mathbb C: |z-a|\leq \frac{1}{2}R\}\), we have

    \[|f(w_1) - f(w_2)| \leq \frac{4M}{R} |w_1 - w_2|.\]

(b) Explain how Part (a) can be used with the Arzela-Ascoli Theorem to prove Montel's Theorem asserting the normality of any locally bounded family \(F\) of analytic functions on a region \(G\).