DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH REAL AND COMPLEX PRELIMINARY EXAMINATION AUGUST 17 2004

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\), and let \(f\ge 0\) be a function which is integrable with respect to \(\mu \). Let \(f_1\le f_2\le \cdots \le f_n\le \cdots \) be a sequence of real valued, measurable functions on \(X\), such that \(f_n\le f\) for each \(n\) and \(\int _X f_n\,d\mu \to \int _X f\,d\mu \). Show that \(f_n\to f\) almost everywhere.

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) such that \(\mu (X)=1\). Let \(f\) be a bounded measurable function on \(X\). Show that \(f\in L^p(X)\) for \(1\le p<\infty \) and \(\|f\|_p\le \|f\|_q\) for \(1<p<q<\infty \).

Let \(u_1,u_2,\cdots ,u_n,\cdots \) be an orthonormal set of vectors in a Hilbert space \(H\). Show that for every \(x\in H\) \[ \lim _{n\to \infty }(x,u_n)=0. \]

Let \(f_1,f_2,\cdots ,f_n,\cdots \) be a sequence of functions in \(L^1(\mathbb R)\) such that \[ \lim _{n\to \infty }\int _{\mathbb R} f_n(x)g(x)\,dx=0 \] for every \(g\in L^\infty (\mathbb R)\). Show that \(f_1,f_2,\cdots ,f_n,\cdots \) is a bounded sequence in \(L^1(\mathbb R)\).

Let \(f,g\in L^1(\mathbb R)\). Let \(A=f^{-1}(\mathbb C-\{0\})\) and \(B=g^{-1}(\mathbb C-\{0\})\). Show that \[ (f*g)^{-1}(\mathbb C-\{0\})\subset A+B \] where \(A+B=\{x+y:x\in A,y\in B\}\).

Let \(f\) be an entire function such that \(f'=f\). Show that \[ f(z)=f(0)e^z=f(0)\sum _{n=0}^{\infty }\frac {z^n}{n!} \] for any \(z\in \mathbb C\).

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

Evaluate the integral \[ \int _0^\infty \frac {\sin ax}{x(x^2+b^2)}\,dx \] where \(a\) and \(b\) are real numbers and \(b\ne 0\).

Let \(f\) be a meromorphic function on the Riemann sphere. Show that \(f\) is a rational function.

Let \(f\) be a holomorphic automorphism of \(\mathbb C^*\). Show that either \(f(z)=\alpha z\) or \(f(z)=\frac {\alpha }{z}\) for some complex number \(\alpha \ne 0\).