91Ó°ÊÓ

Suppose \(f(z) \rightarrow \infty\) as \(z \rightarrow z_{0}\), an isolated singularity. Show that \(f\) has a pole at \(z_{0}\).

Suppose \(f\) and \(g\) are entire functions with \(|f(z)| \leq|g(z)|\) for all \(z\). Prove that \(f(z)=c g(z)\), for some constant \(c\).

Show that if \(f\) is analytic in \(z \neq 0\) and "odd" (i.e., \(f(-z)=-f(z))\) then all the even terms in its Laurent expansion about 0 are \(0 .\)

Suppose \(f\) is analytic in a deleted neighborhood \(D\) of \(z_{0}\) except for poles at all points of a sequence \(\left\\{z_{n}\right\\} \rightarrow z_{0}\). (Note that \(z_{0}\) is not an isolated singularity.) Show that \(f(D)\) is dense in the complex plane. [Hint: Assume, as in the proof of the Casorati- Weierstrass Theorem, that \(|f(z)-w|>\delta\) and consider \(g(z)=1 /(f(z)-w) .]\)