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I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?

user1050421
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  • The body of your Question should be as self-contained as possible. The presentation here is hopelessly terse. Perhaps a true specialist in differential geometry might be able to make a reasonable guess as to your meaning, but additional context would benefit your future Readers as well as those able to Answer. – hardmath Apr 26 '16 at 22:09
  • Please do not edit your question into nothingness. – pjs36 Apr 26 '16 at 23:37

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Hint: $\sin(z)=z\bigl(1+z^2g(z)\bigr)$ where $g$ is analytic with $g(0)\ne0$.

Edit: So $$\frac1z-\frac1{\sin z}=\frac{\sin z-z}{z\sin z}=\frac{z^3g(z)}{z^2\bigl(1+z^2g(z)\bigr)}.$$ Can you take it from there?

Harald Hanche-Olsen
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  • My hint is very precise; can't make it more so. But if you want more hints, try to write the fractions on a common denominator. – Harald Hanche-Olsen Apr 26 '16 at 20:10
  • I am sorry, but I am stuck. Could you tell me more? – user1050421 Apr 26 '16 at 20:16
  • Honnestly, I am able to find $g(z)$, but how it allows me to conclude? I don't know the result that allows me to conclude from there. – user1050421 Apr 26 '16 at 20:44
  • What do you know about the classification of singularities? Essential, poles, removable? Are those familiar terms? Do you know some of the basic properties of each type? – Harald Hanche-Olsen Apr 26 '16 at 20:55
  • I know some properties in using summation or limit, but not only with simple functions. Which properties allow me to conclude here? – user1050421 Apr 26 '16 at 20:56
  • The singularity is removable if and only if the function remains bounded near the singularity. It is a pole if the absolute value goes to infinity as you approach the singularity. If neither hold, it is essential. (I wonder what kind of book gives this sort of exercise without having told the students these basic facts first …) – Harald Hanche-Olsen Apr 26 '16 at 20:59
  • the lecture notes from my teacher... – user1050421 Apr 26 '16 at 21:21
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    What dot you mean by "bound near the singularity"? Could you complete your answer? I have an exam tomorrow and it would help me a lot to understand; I would appreciate! – user1050421 Apr 26 '16 at 21:52
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    I would like also understanding what you did... Could you explain? –  Apr 26 '16 at 22:07