DEPARTMENT OF MATHEMATICS University of Utah Real and Complex Analysis Preliminary Examination January 2, 2013

Let \(f\in L^p(\mathbb R)\) (with respect to the Lebesgue measure), and \(1\le p\le \infty \), and let \[ F(x)=\int _0^x f(t)\,dt. \] Prove that \(F\) defines a continuous function on \(\mathbb R\).

Let \((X,\mathcal M,\mu )\) be a measure space, and let \(f\ge 0\) be an integrable function on \(X\). Suppose that \(f_n\ge 0\) is a measurable function, \(n=1,2,\ldots \), and that \[ f_1\le f_2\le \cdots \le f. \] If \(\int f_n\,d\mu \to \int f\,d\mu \), prove that \(f_n\to f\) a.e.

Let \((X,\mathcal M,\mu )\) be a measure space, \(1<p<\infty \), and \(q=p/(p-1)\). Suppose that \(f_n\in L^p(X)\), \(n=1,2,\ldots \), and that for each \(g\in L^q(X)\), \[ \sup _n \left |\int f_ng\,d\mu \right |<\infty . \] Show that \(\sup _n\lVert f_n\rVert _p<\infty \).

Let \(H\) be a Hilbert space, with \(U\) and \(V\) closed subspaces which are orthogonal. Prove that \(U+V\) is a closed subspace of \(H\).

Consider \(\mathbb R\) with the Lebesgue measure, and recall the definitions of convolution and the Fourier transform, for \(f,g\in L^1(\mathbb R)\): \[ f*g(x)=\int f(x-y)g(y)\,dy,\qquad \hat f(\xi )=\int f(x)e^{-2\pi ix\xi }\,dx. \] Show that for \(f,g\in L^1(\mathbb R)\), \(f*g\in L^1(\mathbb R)\), and that \[ \widehat {f*g}=\hat f\,\hat g. \]

Find the Taylor series of the function \[ f(z)=\frac {z}{2+z} \] at the point \(a=1\). What is the radius of convergence of that series?

Let \(f\) be a holomorphic function and \(a\) an isolated singularity of \(f\). Prove that the following statements are equivalent:

(a) there exists a disk \(D(a,\epsilon )=\{z\in \mathbb C\mid |z-a|<\epsilon \}\) and a constant \(C>0\) such that \[ |f(z)|\ge \frac {C}{|z-a|^n} \] for all \(z\in D(a,\epsilon )-\{a\}\);

(b) \(f\) has a pole of order \(m\ge n\) at \(a\).

Find the Laurent series of the function \[ f(z)=\frac {1}{(z-z^2)^2} \] at \(z=0\).

(a) What kind of isolated singularity does \(f\) have at \(0\)?

(b) What is the residue of \(f\) at \(0\)?

Evaluate the integral \[ \int _0^\infty \frac {x^2+1}{x^4+1}\,dx. \]

Let \(f\) be a holomorphic function in a neighborhood of the closed unit disk centered at \(0\). Assume that \(|f(z)|<1\) for any \(z\) on the unit circle centered at \(0\). How many fixed points does \(f\) have inside this unit circle?