DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH REAL AND COMPLEX ANALYSIS PRELIMINARY EXAMINATION January 3, 2005

Let \(f\) and \(g\) be two real measurable functions on a measurable space \(X\). Show that \[ \{x\in X\mid \sin (f(x))\ge \exp (g(x))\} \] is measurable.

Let \(f_1,f_2,\ldots \) be a sequence of real-valued convex functions on \(\mathbb R\). Define their upper-limit via \[ \overline f(x)=\limsup _{n\to \infty } f_n(x); \] and the lower-limit via \[ \underline f(x)=\liminf _{n\to \infty } f_n(x). \] Prove or find a counterexample to each of the following two statements:

(i) \(\overline f\) is convex.

(ii) \(\underline f\) is convex.

Let \(f\) be a bounded linear functional on a linear subspace \(M\) of a Hilbert space \(H\).

(i) Prove that \(f\) has a unique extension to a bounded linear functional \(F\) on \(H\) so that the norm of \(f\) coincides with the norm of \(F\).

(ii) Prove that the extension \(F\) vanishes on \(M^\perp \).

Let \(H\) be an infinite dimensional Hilbert space. Describe a closed subset of \(H\) which contains no element of minimal norm.

Suppose \(X\) consists of two points \(a\) and \(b\). Define \(\mu (\{a\})=1\), \(\mu (\{b\})=\mu (X)=\infty \), and \(\mu (\emptyset )=0\). For this measure \(\mu \), prove or disprove: \(L^\infty (\mu )\) is the dual space of \(L^1(\mu )\).

Determine if there exists an entire function such that \[ f\left (\frac {1}{n}\right )=f\left (-\frac {1}{n}\right )=\frac {1}{n^3} \] for all \(n\in \mathbb N\).

Let \[ f(z)=\frac {e^{\frac {1}{z-1}}}{e^z-1}. \] Determine all isolated singularities of \(f\) and their type. Find the residues of \(f\) at its poles.

Evaluate the integral \[ \int _0^{2\pi }\frac {\cos ^2 3\phi }{1-2a\cos \phi +a^2}\,d\phi \] where \(a\) is a complex number such that \(|a|<1\).

Let \(f\) be a holomorphic function with isolated singularity at point \(a\). Assume that \(g=\frac {1}{f}\) also has an isolated singularity at \(a\), and that \(a\) is not a removable singularity for either \(f\) or \(g\). Determine what type of isolated singularity \(a\) is for \(f\) and \(g\).

Do you know an example of such a function \(f\)?

Let \(f\) be a function holomorphic in \(\mathbb C-\{0,2\}\). Assume that:

(i) \(0\) and \(2\) are first order poles of \(f\);

(ii) that \(f\) is bounded for \(|z|\ge R\) for sufficiently large \(R\);

(iii) the integral \(\int _\gamma f(z)\,dz\) is equal to:

a) \(2\pi i\) if \(\gamma \) is a positively oriented circle of radius \(1\) centered at \(0\);

b) \(0\) if \(\gamma \) is a positively oriented circle of radius \(3\) centered at \(0\).

Determine the function \(f\).