Department of Mathematics University of Utah Real and Complex Analysis Preliminary Examination 0.5 January 7, 2011

Recall that a topological space is said to be separable if it contains a countable dense set. Also recall that \(\ell ^p(\mathbb Z)\) is the \(L^p\)-space for counting measure on the integers. Prove or disprove the following statements:

(a) \(\ell ^1(\mathbb Z)\) is separable.

(b) \(\ell ^\infty (\mathbb Z)\) is separable.

Let \(f\) be a real-valued function on a measure space \(X\). Prove or disprove the following statements:

(a) If \(f\) is measurable, then so is \(|f|\).

(b) If \(|f|\) is measurable, then so is \(f\).

Let \(f\) be a continuous function on the circle \(\{e^{i\theta }\mid 0\le \theta <2\pi \}\). Let \[ c_n:=\frac {1}{2\pi }\int _0^{2\pi } f(e^{i\theta })e^{-in\theta }\,d\theta . \] Prove or disprove: \[ f(e^{i\theta })=\lim _{N\to \infty }\left (\sum _{n=-N}^{N}c_ne^{in\theta }\right ) \] for all \(\theta \).

Let \(\mathbb R\) be equipped with Lebesgue measure. Suppose \(f\in L^1(\mathbb R)\) is uniformly continuous. Prove that \(\lim _{|x|\to \infty }f(x)=0\).

Let \((X,\mu )\) be a measure space, let \(\{f_n\}\) be a sequence of nonnegative integrable functions, and assume there exists an integrable function \(f\) such that \(f(x)=\lim _{n\to \infty } f_n(x)\) almost everywhere. Further assume \[ \tag {1} \lim _{n\to \infty }\int _X f_n\,d\mu =\int _X f\,d\mu . \] Prove that \[ \lim _{n\to \infty }\int _X |f_n-f|\,d\mu =0. \] Show by example that this need not hold if the assumption (1) is omitted.

Let \(f\) be a holomorphic function defined on a domain \(U\subset \mathbb C\). Let \[ \bar U=\{z\in \mathbb C\mid \bar z\in U\}. \] Consider the function \(g\) on \(\bar U\) defined by \[ g(z)=\overline {f(\bar z)}. \] Equivalently, \(g(z)\) is the complex conjugate of \(f(\bar z)\). Is \(g\) holomorphic or not? Explain your answer!

Let \(f\) be a meromorphic function on \(\mathbb C\) such that there exist \(R,C>0\) and a positive integer \(k\) such that \(|f(z)|\le C|z|^k\) for all \(|z|\ge R\). Prove that \(f\) is a rational function.

Let \(f\) be a holomorphic function on \(U\subset \mathbb C\). Let \(a\in U\) and let \(k\) be a positive integer. Prove that the following statements are equivalent:

(i) \(a\) is a zero of order \(\ge k\) of \(f\);

(ii) there exist \(C,\epsilon >0\) such that \[ |f(z)|\le C|z-a|^k \] for all \(z\) such that \(|z-a|<\epsilon \).

Find all isolated singularities of the function \[ f(z)=\sin (z)\sin \left (\frac {1}{z}\right ) \] in \(\mathbb C\). Determine the residues of \(f\) at these isolated singularities.

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