91Ó°ÊÓ

Determine the singularities and associated residues of a. \(\frac{1}{z^{4}+z^{2}}\) b. \(\cot z\) c. \(\csc z\) d. \(\frac{\exp \left(1 / z^{2}\right)}{z-1}\) e. \(\frac{1}{z^{2}+3 z+2} \quad\) f. \(\sin \frac{1}{z}\) g. \(z e^{3 / z}\) h. \(\frac{1}{a z^{2}+b z+c}, a \neq 0\).

a. Show that Rouche's Theorem remains valid if the condition: \(|f|>|g|\) on \(\gamma\) is replaced by: \(|f| \geq|g|\) and \(f+g \neq 0\) on \(\gamma\) b. Find the number of zeroes of \(z^{5}+2 z^{4}+1\) in the unit disc.

Suppose \(\lambda>1 .\) Show that \(\lambda-z-e^{-z}=0\) has exactly one root (which is a real number) in the right half-plane.