Exercise · ZW-ZERO-AND-POLE-010 · ★★★ Advanced
\(f\) has a zero of order \(3\) and \(g\) a pole of order \(2\) at \(z_0\). What does \(fg\) have at \(z_0\)?
Choose one option.
Show answer
- C. A zero of order \(1\)
Explanation
With \(f=(z-z_0)^3 f_1\) and \(g=\frac{g_1}{(z-z_0)^2}\), the product is \((z-z_0)\,f_1g_1\) with \(f_1g_1\neq0\) at \(z_0\): a zero of order \(3-2=1\).
Review the concept: Zero and pole
A zero of order \(m\) of a holomorphic function \(f\) is a point \(z_0\) where \(f\) vanishes with multiplicity \(m\), with holomorphic quotient \(g\); a pole of order \(m\) is a point where \(1/f\) has such a zero.
\[f(z)=(z-z_0)^m\,g(z),\qquad g(z_0)\neq 0\]
More practice problems: zero and pole
- ZW-ZERO-AND-POLE-001 What is the order of the zero of z^3sin z at z=0?
- ZW-ZERO-AND-POLE-002 What is the order of the zero of 1-cos z at z=0?
- ZW-ZERO-AND-POLE-003 What kind of singularity does (1)/(z^2+1) have at z=i?
- ZW-ZERO-AND-POLE-004 What is the order of the pole of (1)/(sin ^3 z) at z=0?
- ZW-ZERO-AND-POLE-005 What kind of singularity does (z)/(sin z) have at z=0?
- ZW-ZERO-AND-POLE-006 What is the order of the pole of (z^2-1)/((z-1)^3) at z=1?
- ZW-ZERO-AND-POLE-007 What is the order of the zero of e^z-1-z at z=0?
- ZW-ZERO-AND-POLE-008 What is the order of the pole of (1)/(z (e^z-1)) at z=0?