Department of Mathematics University of Utah Real and Complex Analysis Preliminary Examination 0.5 August 18, 2010

Suppose \((X,\mathcal M,\mu )\) is a measure space and \(f:X\to \mathbb R\) is a real-valued function on \(X\). Suppose further that \(E_r:=\{x\mid f(x)>r\}\) is measureable for each rational number \(r\). Either prove the following assertion or find a counter-example: \(f\) is measureable.

Suppose \((X,\mathcal M,\mu )\) is a measure space and fix \(p\) and \(q\) finite such that \[ \frac {1}{p}+\frac {1}{q}=1. \] Let \(f_1,f_2,\ldots \) be a sequence of functions in \(L^p(X)\) converging (in \(L^p\)) to \(f\), and let \(g_1,g_2,\ldots \) be a sequence of functions in \(L^q\) converging (in \(L^q\)) to \(g\). Prove that the sequence \(f_1g_1,f_2g_2,\ldots \) converges to \(fg\) in \(L^1\). Does the same conclusion hold if \(p=1\) and \(q=\infty \)?

Let \(H\) be a Hilbert space and suppose that \(\{x_n\}\) is a sequence in \(H\) with the following property: for each \(y\in H\), \[ \sup _n |\langle x_n,y\rangle |<\infty . \] Prove that \[ \sup _n \lVert x_n\rVert <\infty . \]

Suppose \(1<p<q<r<\infty \). (Here \(p\) and \(q\) are arbitrary, not necessarily conjugate.) Prove that \(L^p(\mathbb R)\cap L^r(\mathbb R)\subset L^q(\mathbb R)\).

Let \(H\) be a Hilbert space, \(M\) a closed subspace of \(H\), and \(x\in H\). Prove that there is a unique point \(y\in M\) which is closest to \(x\).

Let \[ f(z)=1-\cos z. \] (i) Find all zeros of this function;

(ii) Find the multiplicities of these zeros.

Let \[ f(z)=\sin \left (\frac {z}{z+1}\right ). \] (i) Determine all isolated singularities of \(f\) and their type;

(ii) Find the Laurent expansions of \(f\) at these singularities;

(iii) Find the residues of \(f\) at these singularities.

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {x\cos x}{x^2-2x+10}\,dx \] using the residue theorem.

Let \(n\) be a positive integer. Denote by \(V_n\) the linear space of all entire functions \(f\) such that there exists \(C>0\) such that \(|f(z)|\le C|z|^n\) for all \(z\in \mathbb C\).

(i) Describe precisely the functions in \(V_n\);

(ii) Find the dimension of \(V_n\).

Using Rouché’s theorem find the number of zeros of the polynomial \(2z^5-z^3+3z^2-z+8\) in the region \(\{z\mid |z|>1\}\).