6220-2021-05

Show that any function which is meromorphic on the Riemann sphere, \(\widehat {\mathbb {C}}=\mathbb {C}\cup \{\infty \}\), is either constant or can be written as the sum of the singular parts of its Laurent series at its poles, in particular conclude that the function is rational.

Let \(U\) be a domain in \(\mathbb {C}\). Consider the following collection of functions \[ X=\{f:U\to \mathbb {C}: f\text { holomorphic and }\int _U |f|^2\le M\} \]

where \(\int _U\) is the standard Lebesgue area integral. Show that \(X\) is compact with respect to the topology of uniform convergence on compact subsets of \(U\) (Note: this topology is metrizable so you can show sequential compact- ness).

Show that if \(f(z)\) is entire and nowhere zero then there is an entire function \(g(z)\) such that \(f(z)=g(z)^2\).

Suppose that \(f_n\) are holomorphic on a domain \(U\subset \mathbb {C}\) and \(f_n(U)\subset \mathbb {C}\setminus \{w\}\) for all \(n\) and some fixed \(w\in \mathbb {C}\). Show that if \(f_n\to f\) locally uniformly in \(U\) then either \(f(z)\equiv w\) or \(f(U)\subset \mathbb {C}\setminus \{w\}\).

Suppose that \(f\) is entire and \(f(\mathbb {C})\subset \mathbb {C}\setminus [0,\infty )\), show that \(f\) is constant. Do not use either Picard Theorem (at least not without proof).

Compute the following integral using the techniques of complex analysis \[ \int _0^\infty \frac {\log (x)}{2+x^2}\,dx. \]

Suppose that \(f\) is holomorphic on the upper half plane \(\mathbb {H}=\{\operatorname {Im}(z)>0\}\), \(f\) extends continuously to \(\overline {\mathbb {H}}=\mathbb {H}\cup \mathbb {R}\), and \(f(\mathbb {R})\subset \mathbb {R}\). Show that \(f\) has a unique holomorphic extension to \(\mathbb {C}\) and explicitly identify that extension (in terms of \(f\)).

How many zeros does the function

have in the right half plane \(\{\operatorname {Re}(z)>0\}\).