Department of Mathematics University of Utah Real and Complex Analysis Preliminary Examination

For any subset \(A\) of \(\mathbb R\) we define its Lebesgue outer measure \[ \lambda ^*(A)=\inf \left \{ \sum _{i=0}^{\infty }|b_i-a_i| \;\middle |\; A\subset \bigcup _{i=0}^{\infty }(a_i,b_i) \right \}. \] Recall that given an outer measure \(\phi ^*:\mathcal P(X)\to [0,\infty ]\), we say a set \(A\) is \(\phi ^*\)-measurable if for all \(Y\in \mathcal P(X)\) we have \[ \phi ^*(Y)=\phi ^*(Y\cap A)+\phi ^*(Y\setminus A). \] Show that if \(\lambda ^*(\partial E)=0\), then \(E\) is \(\lambda ^*\)-measurable.

A family of complex-valued functions \(\mathcal S\) on \([0,1]\) is uniformly integrable if for all \(\epsilon >0\) there exists \(\delta >0\) so that \(\lambda (A)<\delta \) implies \(\left |\int _A f\,d\lambda \right |<\epsilon \) for all \(f\in \mathcal S\).

Let \((f_i;i\in \mathbb N)\) be a uniformly integrable sequence. Assume that \[ \lim _{i\to \infty }f_i(x)=g(x)\quad \text {for all }x\in [0,1]. \] Prove that \[ \lim _{i\to \infty }\int _0^1 f_i\,d\lambda =\int _0^1 g\,d\lambda . \]

Let \(f:\mathbb R^2\to \mathbb C\) be a Borel measurable function. Let \(g:\mathbb R^2\to \mathbb R^2\) be the map given by \(g(x,y)=(2x+y^2,y)\) for \((x,y)\in \mathbb R^2\). If \(f\in L^1(\lambda ^2)\), what is \[ \frac {\int _{\mathbb R^2}(f\circ g)\,d\lambda ^2} {\int _{\mathbb R^2}f\,d\lambda ^2}? \]

Let \(\mathcal H\) be a Hilbert space. Prove every orthonormal set is contained in an orthonormal basis.

Consider \(\ell ^4(\mathbb N)\) equipped with topology induced by the norm \(\lVert \cdot \rVert _4\). Show that there exists an open set in this topology that is not open in the weak topology on \(\ell ^4(\mathbb N)\).

What are the possible ranges of holomorphic functions from \(\mathbb C\setminus \{\pi \}\) to \(\mathbb C\)?

Let \(\mathbb D\) be the unit disk \(\{z\in \mathbb C\mid |z|<1\}\). Let \(f\) be a holomorphic function on \(\mathbb D\setminus \{0\}\) such that \[ |f(z)|\le \log \left (\frac {1}{|z|}\right ) \quad \text {for all }z\in \mathbb D\setminus \{0\}. \] Show that \(f=0\).

Let \(f\) be a holomorphic function on the unit disk \(\mathbb D\). Then there exists \(n\in \mathbb N\) so that \[ f\left (\frac {1}{n}\right )\ne \frac {1}{n+2}. \]

Let \(f:U\to \mathbb C\) be a holomorphic function, where \(U\) is an open set. Let \(a\in U\). A standard result in complex analysis is that if \(f'(a)\ne 0\), then there is an open neighborhood \(V\subset U\) of \(a\) on which \(f\) is injective and the image \(f(V)\) is a neighborhood of \(f(a)\). Prove this result!

Using the residue theorem, find the integral \[ \int _{-\infty }^{\infty }\frac {\sin x}{x^2+4x+20}\,dx. \]