There is a theorem that says rational functions in the extended complex plane are exactly the meromorphic functions.After this, my textbook draws the corollary: "...as a consequence, a rational function is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles."I don't see how it follows. Plus I think I have an example. Take a rational function with no zeros and two poles of multiplicity one, at 0 and 1.At least two such functions exist: f ( x ) = 1 x + 1 x − 1 g ( x ) = 1 x + 2 x − 1

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There is a theorem that says rational functions in the extended complex plane are exactly the meromorphic functions.After this, my textbook draws the corollary: &quot;...as a consequence, a rational function is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles.&quot;I don&#039;t see how it follows. Plus I think I have an example. Take a rational function with no zeros and two poles of multiplicity one, at 0 and 1.At least two such functions exist:  f  (  x  )  =      1    x    +      1          x      −      1        g  (  x  )  =      1    x    +      2          x      −      1

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The precise statement is the other way round: a function meromorphic on the whole Riemann sphere   S is rational.This is completely false on       C  : the function       e    z   is meromorphic (even holomorphic) on       C  , but certainly not rational on       C  , nor anywhere else.Now if   f  ,  g  ∈  R  a  t  (  S  ) are rational on the Riemann sphere   S and have the same zeros and poles (counted with multiplicities), then the quotient   ϕ  =  f      /    g having neither zeros nor poles is holomorphic on the whole of   S and thus constant (say because ϕ is bounded on   S by compacity, hence bounded on       C   and Liouville&#039;s theorem applies).&quot;Yes, but I gave you a counterexample&quot;Your rational functions don&#039;t have the same zeros:  f  (      1    2    )  =  0  ≠  g  (      1    2    )  =  −  2A very optional remarkThat meromorphic functions on   S are rational is the simplest example of a very profound result in algebraic geometry: Serre&#039;s GAGA principle.

There is a theorem that says rational functions in the extended complex plane are exactly the meromo

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