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.

  • $\begingroup$ 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

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 $2$.

  • $\begingroup$ 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

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