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There is a theorem which says

The meromorphic functions in the extended complex plane are the rational functions.

Right after the proof, it says

Note that as a consequence, a rational [emphasis mine] function is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeroes and poles.

My question is, should't the bold word be replaced by "meromorphic"? We do not need the above theorem to conclude that a rational function is determined by that information. It would make more sense if the book said that a meromorphic function on the extended plane is determined by that information.

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    $\begingroup$ But a function meromorphic on the extended plane is rational, as you have said. The two sentences mean exactly the same thing. Why does the second make more sense? $\endgroup$ – saulspatz Mar 3 at 23:59
  • $\begingroup$ Isn't it meromorphic functions (not rational functions) that we usually speak of as having poles? But now since the two are identical... $\endgroup$ – Chris Custer Mar 4 at 0:13
  • $\begingroup$ @saulspatz Because the first is obvious without the theorem; and the theorem doesn’t seem to be involved at all. The second sentence is only true in light of the theorem (and they use the “consequently”, which makes you expect a result that is true in light of the theorem, not one that doesn’t really have anything to do with the theorem) $\endgroup$ – Ovi Mar 4 at 2:21
  • $\begingroup$ @ChrisCuster Please see my comment above. $\endgroup$ – Ovi Mar 4 at 2:21
  • $\begingroup$ @ChrisCuster $\tan{z}$ is not meromorphic on the entire Riemann sphere; it has an essential singularity at $\infty$ $\endgroup$ – saulspatz Mar 4 at 5:17
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You are right. Theorem 3.4 shows that the meromorphic functions in the extended complex plane agree with the rational functions.

Moreover it is clear that a rational function is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles.

Hence the same holds for a meromorphic function, but this was not clear before Theorem 3.4 was proved.

The statement after "as a consequence" is certainly a little inattentiveness of the authors.

Finally note that it is essential to consider meromorphic functions in the extended complex plane. For a meromorphic functions in the complex plane the assertion of Theorem 3.4. is false. A rational function in the complex plane is of course meromorphic and it is moreover determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles. However, the are more meromorphic functions than rational functions. As an example take the functions $\sin z$ and $e^z \sin z$ which are entire (hence meromorphic) but are not rational. Both have the same set of zeros with multplicity $1$ and no poles.

The problem is that each meromorphic function in the extended complex plane restricts to a meromorphic function in the complex plane, but not each meromorphic function in the complex plane is obtained as such a restriction. In fact, if $f$ is meromorphic the complex plane, then $\infty$ may be an essential singularity $f$ or it may be a cluster point of zeros or poles of $f$. Both situations prevent to consider $f$ as a meromorphic function in the extended complex plane.

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