MAT334-2018F > Quiz-6

Q6 TUT 5101

(1/2) > >>

Victor Ivrii:
Locate each of the isolated singularities of the given function $f(z)$ and tell whether it is a removable singularity, a pole, or an essential singularity.

If the singularity is removable, give the value of the function at the point; if the singularity is a pole, give the order of the pole:
$$f(z)= \frac{e^z-1}{e^{2z}-1}.$$

Yatong Yu:
let $f(z)=e^z-1$  $g(z)=e^{2z}-1$

$e^{2z}-1=0$ so $e^{2z}=1$  $e^z=\pm1$

when  $e^z=+1$
$f(z)=e^z-1=0$ $f'(z)=e^z=1\neq0$    so order=1
$g(z)=e^{2z}-1=0$ $g'(z)=2e^{2z}=2\neq0$    so order=1
1-1=0   removable

when  $e^z=-1$
$f(z)=e^z-1=-2\neq0$    so order=0
$g(z)=e^{2z}-1=0$ $g'(z)=2e^{2z}=2\neq0$    so order=1
1-0=1   simple pole

Zihan Wan:
solution is attached

Qing Zong:
The value at the point should be 1/2

Qing Zong:
This is extra step for 1/2