0
$\begingroup$

i would like to know how to determine if pole of given function is inside a circle of radius 2? for example let us take this function $$ f(z)=1/\cos z $$ We have poles at $$ z=\pm \pi/2, \pm 3\pi/2, \pm 5\pi/2, \dotsc $$ So which of these poles should be considered for a circle whose radius is 2.

$\endgroup$
  • $\begingroup$ the ones in the circle: $\pm \pi/2$ $\endgroup$ – james1395 Apr 28 '16 at 10:53
  • $\begingroup$ That depends on where the center of the circle is. $\endgroup$ – Henning Makholm Apr 28 '16 at 11:39
1
$\begingroup$

You only have to compute the modules and see if they are less than $2$.

For exemple $|\pm \frac{\pi}{2}|<2$ since it is well known that $3<\pi<3,5$.

However the other singularities are not in the disk because the module is greater than $2$.

$\endgroup$
  • $\begingroup$ It means if the contour is unit circle then all the singularities lies outside the circle and I can directly take the integral of the function equal to zero by Cauchy's residue theorem. $\endgroup$ – fahad shaikh Apr 28 '16 at 11:07
  • $\begingroup$ This is correct. $\endgroup$ – C. Dubussy Apr 28 '16 at 11:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.