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I have to provide a function $f(z)$ which is analytic in C except for (i) at a single pole at $z = 3$, (ii) a removable singularity at $z=0$ and (iii) an essential singularity at $z=-1$.

I've defined $f(z) = f_i +f_{ii} + f_{iii}$.

Where $f_{i}(z) = \frac{z+1}{z+3}$, $f_{ii}(z) = \frac{1}{z}$, and $f_{iii}(z) = sin\left(\frac{7}{z+1}\right)$.

I have referred to this site.

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  • $\begingroup$ What makes you think that $\frac{1}{z}$ has a removable singularity at $z=0$? $\endgroup$ – almagest Apr 20 '16 at 17:52
  • $\begingroup$ This $f$ has a pole at $z=0$. $\endgroup$ – Rick Sanchez Apr 20 '16 at 17:52
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How about $$f(z)=\frac{z+1}{z-3} +\frac{\sin z}{z}+e^{\frac{1}{z+1}}.$$

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    $\begingroup$ I was reading this wikipedia document when you posted an answer. $\endgroup$ – Bayerischer Apr 20 '16 at 17:58
  • $\begingroup$ Why did you also change my last term of my proposed equation? $\endgroup$ – Bayerischer Apr 20 '16 at 17:58
  • $\begingroup$ Your last one is fine. $\endgroup$ – Rick Sanchez Apr 20 '16 at 17:59

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