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

If \(f\in L^1(\lambda )\) then \[ \lim _{c\to 1}\int |f(cx)-f(x)|\,d\lambda =0. \]

If \(f:\mathbb R\to \mathbb R\) is Lipschitz and \(A\) is \(\lambda \)-measurable, then \(f(A)\) is measurable. Recall that a function is Lipschitz if there exists \(C\) so that \[ d(f(x),f(y))\le C\,d(x,y) \] for all \(x,y\in \mathbb R\).

Show that if \(f\in L^1\), then \[ \lim _{x\to \infty }\mathcal F(f)(x)=0. \]

Show that \[ \left \{\bar v\in \ell ^p:\ |v_i|\le 2^{-i}\right \} \] is compact for all \(p\ge 1\).

Prove that if \(\mu \) is a non-atomic Borel probability measure on \(\mathbb R\), then the function \(f:\mathbb R\to \mathbb R\) defined by \[ f(t)=\mu ((-\infty ,t)) \] is continuous.

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x\sin x}{x^2+a^2}\,dx, \] for \(a\in \mathbb R\).

Let \(f(z)\) be analytic in a deleted neighborhood of \(z=a\) and \[ \lim _{z\to a}(z-a)f(z)=0. \] Does it mean that \(f(z)\) has a removable singularity at \(a\)? If so, prove it. If not, give a counterexample.

If \(f(z)\) is analytic in \(|z|\le 1\) and satisfies \(|f|=1\) on \(|z|=1\), show that \(f(z)\) is rational.

Prove that in any region \(\Omega \) the family of analytic functions whose values lie on an open half-plane is normal.

Let \(\wp (z)\) be the Weierstrass \(\wp \) function. Show that any even elliptic function \(f(z)\) with periods \(\omega _1,\omega _2\) can be expressed in the form \[ (\text {Constant})\prod _{k=1}^{n} \frac {\wp (z)-\wp (a_k)}{\wp (z)-\wp (b_k)} \] for some Constant, \(a_k,b_k\in \mathbb C\), provided that \(0\) is neither a zero nor a pole. What is the corresponding form if \(f(z)\) either vanishes or has poles at \(z=0\)?