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Function $f(z)$ is an entire function such that $$|f(z)| \le |z^{n}|$$ for $z \in \mathbb{C}$ and some $n \in \mathbb{N}$.

Show that the singularities of the function $$\frac {f(z)}{z^{n}}$$ are removable. What can be implied about the function $f(z)$ if moreover $f(1) = i$? Draw a far-reaching conclusion.

My attempt: If the singularities of $\frac {f(z)}{z^{n}}$ are removable, it is entire (not sure, need help with the justification) and bounded, so constant from Liouville's theorem, the constant value of the funcion is $i$, hence $f(z)=iz^{n}$.

But what about the $n$ here, is it arbitrary? Could somebody help me prove the removability of the singularities and suggest if my attempt is going the right way?

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  • $\begingroup$ What kind of singularities exist and how are they distinguishable? $\endgroup$ – Hagen von Eitzen Jun 27 '17 at 13:51
  • $\begingroup$ @Theta, you read about singularities in holomorphic functions. They are of 3 kinds. I have provided a solution to your question, but I advise you to read about them in your textbook. $\endgroup$ – user456218 Jun 27 '17 at 14:00
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Since $f(z)$ is an entire function, $g(z)=\frac{f(z)}{z^n}$ may only have a pole at the origin. If that was the case, then $\left|g(z)\right|$ would be unbounded in a neighbourhood of the origin, but that contradicts $\left|g(z)\right|\leq 1$ for any $z\neq 0$. It follows that $z=0$ is a removable singularity for $g(z)$ and $g(z)$ is an entire function. Since $g(z)$ is bounded, by Liouville's theorem it follows that $g(z)$ is constant, so $f(z)=C z^n$. If $f(1)=i$, it follows that $f(z)=i z^n$.

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    $\begingroup$ @Theta: $f(z)$ can be expanded as a Taylor series in a neighbourhood of zero, $f(z)=a_0+a_1 z+a_2 z^2+\ldots$. It follows that the Laurent expansion of $g(z)$ is given by $$ \frac{a_0}{z^n}+\frac{a_1}{z^{n-1}}+\ldots+ a_n + a_{n+1} z +\ldots . $$ If some coefficient among $\{a_0,\ldots,a_{n-1}\}$ were different from zero, what $\lim_{z\to 0}|g(z)|$ would be? $\endgroup$ – Jack D'Aurizio Jun 27 '17 at 15:14
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    $\begingroup$ It would become infinitely large as $z$'s getting closer and closer to $0$. Thank you, now I've developed some image of the pole behaviour in my mind. :) And if the point's a removable singularity, the coefficients you mentioned are all zeros, thus we can substitute $z=0$ and $a_{n}$'s a finite limit. Am I right that boundedness follows from the maximum modulus principle? $\endgroup$ – Theta Jun 27 '17 at 15:34
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    $\begingroup$ @Theta: absolutely correct. $\endgroup$ – Jack D'Aurizio Jun 27 '17 at 15:41
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    $\begingroup$ @Theta:in order to apply the MMP you have to prove that $g(z)$ is a holomorphic function, hence yes, to prove that $z=0$ is a removable discontinuity is crucial. $\endgroup$ – Jack D'Aurizio Jun 27 '17 at 15:50
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    $\begingroup$ Thank you very much, so many kind and knowledgeable people here, and complex analysis is an immensely interesting topic. :) $\endgroup$ – Theta Jun 27 '17 at 15:55
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Consider $\frac{f(z)}{z^n}$. Since $f$ was entire, we just need to show that the singularity at $0$ is removable. Because of Reimann's theorem(read here https://en.wikipedia.org/wiki/Removable_singularity#Riemann.27s_theorem) this is equivalent to showing that

$$\lim_ {z \to 0}z\frac{f(z)}{z^n} = \lim_ {z \to 0}\frac{f(z)}{z^{n-1}} =0 $$ But this follows from your information that $|f(z)| \leq |z|^n$.

Rest of your proof was correct. As far as $n$ is concerned, it will be equal to your earlier $n$.(Prove this !)

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