Exercise · ZW-ZERO-AND-POLE-006 · ★★☆ Intermediate
What is the order of the pole of \(\frac{z^2-1}{(z-1)^3}\) at \(z=1\)?
Choose one option.
Show answer
- B. \(2\)
Explanation
Since \(z^2-1=(z-1)(z+1)\), the function is \(\frac{z+1}{(z-1)^2}\) and the numerator does not vanish at \(1\): the pole has order \(2\).
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-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?
- ZW-ZERO-AND-POLE-009 f has a zero of order 2 and g one of order 5 at z_0. What does f/g have at z_0?