Department of Mathematics University of Utah Written Qualifying Exam in Real and Complex Analysis Fall 1994 Instructions

Let \(\mathcal F=\{f_\alpha \}_{\alpha \in A}\) be a family of continuous complex valued functions on a compact Hausdorff space. Let \(\{c_\alpha \}_{\alpha \in A}\) be a family of complex numbers. Suppose that for each \(F\subset A\) with \(F\) finite, there exists a complex measure \(\mu _F\) of total variation one on \(X\) such that \[ \int _X f_\alpha \,d\mu _F=c_\alpha \qquad (\forall \alpha \in F). \] Show that there exists a complex measure \(\mu \) on \(X\) of total variation at most one such that \[ \int _X f_\alpha \,d\mu =c_\alpha \qquad (\forall \alpha \in A). \]

Let \((X,\mathfrak M,\mu )\) be a positive measure space. Let \(\psi \in L^2(\mu )\cap L^\infty (\mu )\). Show that \(\psi \in L^p(\mu )\) for all \(p\ge 2\).

Suppose that \((X,\mathfrak M,\mu )\) is a measure space with \(\mu \) a finite positive measure. Let \(I\) be an open interval in \(\mathbb R\) and \(f:X\times I\to \mathbb C\) a bounded function such that \(x\to f(x,t)\) is \(\mathfrak M\)-measurable for each \(t\in I\) and \(t\to f(x,t)\) is differentiable for each \(x\in X\). If \(\frac {d}{dt}f(x,t)\) is also bounded on \(X\times I\) prove that \(h(t)=\int _X f(x,t)\,d\mu (x)\) is differentiable on \(I\) and in fact \[ h'(t)=\int _X \frac {d}{dt}f(x,t)\,d\mu (x). \]

Let \(A\), \(B\), and \(C\) be Banach spaces and let \(f:A\to C\) and \(g:B\to C\) be bounded linear maps. Suppose that \(g\) is one to one and \(f(A)\subset g(B)\). Prove that there is a bounded linear map \(h:A\to B\) such that \(b=h(a)\) if and only if \(f(a)=g(b)\).

Prove that if \(A\) and \(B\) are closed subspaces of a Hilbert space \(H\), \(A\perp B\), then \(A+B\) is also closed in \(H\).

Let \(f(z)=\sum _{n=0}^{\infty }a_nz^n\). Assume that \(a_0=0\) and \(a_1=1\). Prove that \(f\) is one to one on the unit disk \(U=\{z\in \mathbb C:|z|<1\}\) if \(\sum _{\ell =2}^{\infty }|a_\ell |\le 1\). (Hint: Show that for every \(z_0\in U\) the function \(z\to f(z)-f(z_0)=z-z_0+\sum _{n=2}^{\infty }a_n(z^n-z_0^n)\) has a simple zero in \(U\).)

Let \(P(z)=a_nz^n+\cdots +a_0\) and suppose that \(P\) has zeroes at \(z_1,\ldots ,z_r\) each with multiplicity \(m_1,\ldots ,m_r\) respectively. Let \(\Omega \) be some domain not containing \(z_1,\ldots ,z_r\), and \(\gamma \) a cycle in \(\Omega \). Suppose further that there exists \(g\) holomorphic on \(\Omega \) such that \(g^p=P\). Show that there exists \(k\in \mathbb Z\) such that \[ m_1\operatorname {Ind}(\gamma ,z_1)+\cdots +m_r\operatorname {Ind}(\gamma ,z_r)=kp. \]

Let \(f\) be an entire function with the property that \(f^{-1}(B)\) is bounded or empty for every \(B\subset \mathbb C\) bounded. Prove that \(f(\mathbb C)=\mathbb C\).

Suppose that \(f\) and \(g\) are one to one analytic functions of the open unit disk \(U=\{z\in \mathbb C:|z|<1\}\) onto some domain \(\Omega \). Suppose that \(f(0)=g(0)\), and \(f'(0)=g'(0)\ne 0\). Show that \(f(z)=g(z)\).

Let \(f\) be holomorphic in an open set containing the disk \(D=\{z\in \mathbb C:|z|\le r\}\) and suppose that the only zero of \(f\) in \(D\) is a simple zero at \(a\) belonging to the interior of \(D\). If \(C_r\) is the positively oriented contour \(|z|=r\) show that \[ \int _{C_r}\frac {1}{f(z)}\,dz=\frac {2\pi i}{f'(a)}. \]

Use methods of complex analysis to find \[ \int _{-\infty }^{\infty }\frac {dx}{x^4+1}. \]