Let F ( z ) be a rational function P ( z ) Q ( z ) such that the degree of P ( z ) is less than the degree of Q ( z ) and suppose that all the zeros of Q ( z ) are contained in the open disk | z | &lt; r.I know that if f ( z ) is analytic for | z | &gt; r and bounded by M &gt; 0 there, that is, | f ( z ) | ≤ M for all z with | z | &gt; r, then the coefficients of the Laurent series of f ( z ) for | z | &gt; r satisfy a j = 0 for j = 1 , 2 , 3 ,....I'm supposed to show that the coefficients of the Laurent series of f ( z ) for | z | &gt; r satisfy a j = 0 for j = 1 , 2 , 3 ,.... by using the corollary above. I know that F ( z ) is analytic for | z | &gt; r but I'm missing the boundless condition to finish my proof. Any suggestions?

Question

Let   F  (  z  ) be a rational function             P      (      z      )              Q      (      z      )       such that the degree of   P  (  z  ) is less than the degree of   Q  (  z  ) and suppose that all the zeros of   Q  (  z  ) are contained in the open disk       |    z      |    &amp;lt;  r.I know that if   f  (  z  ) is analytic for       |    z      |    &amp;gt;  r and bounded by   M  &amp;gt;  0 there, that is,       |    f  (  z  )      |        ≤    M for all   z with       |    z      |    &amp;gt;  r, then the coefficients of the Laurent series of   f  (  z  ) for       |    z      |    &amp;gt;  r satisfy   a  j  =  0 for   j  =  1  ,  2  ,  3  ,....I&#039;m supposed to show that the coefficients of the Laurent series of   f  (  z  ) for       |    z      |    &amp;gt;  r satisfy   a  j  =  0 for   j  =  1  ,  2  ,  3  ,.... by using the corollary above. I know that   F  (  z  ) is analytic for       |    z      |    &amp;gt;  r but I&#039;m missing the boundless condition to finish my proof. Any suggestions?

Blaine Foster · Accepted Answer

As with any nonconstant polynomial,       |    Q  (  z  )      |    →  ∞ as       |    z      |    →  ∞  .. Then   deg  ⁡  (  P  )  &amp;lt;  deg  ⁡  (  Q  ) implies       |    F  (  z  )      |    →  0 as       |    z      |    →  ∞ and in particular is bounded for       |    z      |    &amp;gt;  r, say by   M. The Laurent series of   F centered at some       z    0   has coefficients      a    k    =      1          2      π      i            ∫                  |            z      −              z        0                    |            =      R                  F      (      z      )              (      z      −              z        0                    )                  k          +          1                      d  z  ,so      |        a    k        |    ≤      M          R      k      for big enough   R. If   k  ≥  1, you can take   R  →  ∞ to show       a    k    =  0..

Let F ( z ) be a rational function P ( z )

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