Define f : D ( 0 , 1 ) − &gt; C to be holomorphic such that f ( 0 ) = 0. I want to extend f ( z ) z to be continuous on D ¯ ( 0 , r ) for arbitrary 0 &lt; r &lt; 1.My initial guess was to define: g ( z ) = f ( z ) z for non-zero z but extend g ( z ) to be 0 for z = 0.But if I take lim z → 0 f ( z ) z = f ′ ( 0 ) using L'Hopital's rule. So I think this should be the expression for g ( 0 )? I wonder if:i) L'hopital's rule is applicable to complex functions, and,ii)Am I taking the limit incorrectly? I also do not see a use for f ( 0 ) = 0 in my deduction so I'm rather perplexed.Any help would be appreciated!

Question

Define   f  :  D  (  0  ,  1  )  −  &amp;gt;  C to be holomorphic such that   f  (  0  )  =  0. I want to extend             f      (      z      )        z   to be continuous on       D    ¯    (  0  ,  r  ) for arbitrary   0  &amp;lt;  r  &amp;lt;  1.My initial guess was to define:  g  (  z  )  =            f      (      z      )        z   for non-zero   z but extend   g  (  z  ) to be 0 for   z  =  0.But if I take       lim          z      →      0                  f      (      z      )        z    =      f    ′    (  0  ) using L&#039;Hopital&#039;s rule. So I think this should be the expression for   g  (  0  )? I wonder if:i) L&#039;hopital&#039;s rule is applicable to complex functions, and,ii)Am I taking the limit incorrectly? I also do not see a use for   f  (  0  )  =  0 in my deduction so I&#039;m rather perplexed.Any help would be appreciated!

yelashwag8 · Accepted Answer

If   f  (  0  )  ≠  0, then the limit       lim          z      →      0                  f      (      z      )        z   does not exist (in       C  ). On the other hand, if   f  (  0  )  =  0, then       lim          z      →      0                  f      (      z      )        z    =      f    ′    (  0  ) byt the definition of       f    ′  . So, extend   f to                    F        :                    D        (        0        ,        1        )                    ⟶                              C                                          z                    ↦                              {                                                                                          f                    (                    z                    )                                    z                                                             if                 z                ≠                0                                                                                      f                  ′                                (                0                )                                             if                 z                =                0                ,                                                                  and then   F is continuous. In particular, if   r  ∈  (  0  ,  1  ), the restriction of   F to             D      (      0      ,      r      )        ¯   is continuous.

Define f:D(0,1)−>C to be holomorphic such thatf(0)=0. I want to extend (f(z))/z to be continuous on overline(D)(0,r) for arbitrary 0<r<1.

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