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I am trying to find and classify the singularities of the following :

$$\frac{1}{\cos(Z) + \sin(Z)}$$

and I am not sure how to approach it, I understand the denominator must be zero but I am not sure when that occurs and thus how to classify the singularities.

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  • $\begingroup$ please use mathjax in the future. $\endgroup$ Commented Apr 9, 2020 at 10:24

1 Answer 1

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If $\cos z=0$ the $\cos z+\sin z$ cannot be $0$. If $\cos z \neq 0$ then $\cos z+\sin =0$ iff $\tan z = -1$ which gives $z=-\frac {\pi} 4+2n \pi $ where $n$ is an integer or $z=\frac {3\pi} 4+2n \pi $ where $n$ is an integer.

At each of these points $z_0$ we have $\frac {z-z_0} {\cos z+\sin z} \to \frac 1 {2\cos z_0}$ so these points are simple poles.

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  • $\begingroup$ Thank you! Would this be classified as an essential singularity? $\endgroup$
    – user6789
    Commented Apr 9, 2020 at 10:43
  • $\begingroup$ @user6789 They are simple poles. I have edited my answer to indicate this. $\endgroup$ Commented Apr 9, 2020 at 11:41

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