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Sequences of complex functions

Complex Analysis
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Grigorios Kostakos
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Sequences of complex functions

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Post by Grigorios Kostakos » Wed Apr 19, 2017 7:23 am

In the following cases examine whether the sequence $\{f_n\}_{n\in\mathbb{N}}$ of complex functions converges uniformly or not:
  1. $f_n(z)=z^n\,(1-i\,z)^n\,, \quad |z|<1$,
  2. $f_n(z)=\dfrac{{\rm{e}}^{-n\,\Re(z)\,i}}{n\,|z|}\,, \quad z\in\mathbb{C}\setminus\{0\}$,
  3. $f_n(z)={\rm{Arg}}\,\big(\frac{z}{n\,\overline{z}}\big)\,,\quad z^2+|z|^2\neq0$.
Grigorios Kostakos
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