All Questions
6
questions
1
vote
1
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256
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Continuity of maximum modulus function $M(r)=\max_{|z|=r}|f(z)|$
I am looking to prove that the maximum modulus function
$$M(r)=\max_{|z|=r}|f(z)|$$
is continuous on $[0, \infty)$ for $f$ an entire function.
My idea was to use the representation of $f$ as a power ...
- 5,611
2
votes
1
answer
218
views
$f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.
Question: Suppose we have functions $f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.
My attempt: Consider function $h(z)=\frac{f(z)}{g(z)}$, ...
- 2,534
1
vote
0
answers
81
views
Prove the existence of an entire function satisfying these conditions
This question was asked in an assignment that I am trying to solve.
Let $f:\mathbb{R} \to \mathbb{C}$ be continuous. Then prove the existence of an entire function $g$ such that $|f(x)-g(x)|< 1$ ...
user775699
1
vote
2
answers
112
views
Is $f(z)$ entire?
I am trying to determine if the the following is entire $$f(z)= \begin{cases}
e^{-z^{-4}} & z\neq0 \\
0 & z=0\\
\end{cases}
$$
My attempt:
Consider $z\ne 0$. $f(z)=e^{-z^{-4}...
user557493
6
votes
1
answer
3k
views
If $f: \mathbb{C} \to \mathbb{C}$ is continuous and analytic off $[-1,1]$ then is entire.
This is a problem from Complex Variable (Conway's book) 2nd ed.
(Section 4.4) 9. Show that if $f: \mathbb{C}\to\mathbb{C}$ is a continuous function such that $f$ is analytic off $[-1,1]$ then $f$ is ...
- 2,505
4
votes
1
answer
505
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If $(f(z))^2$ and $(f(z))^3$ are entire functions ; then is $f$ entire? [duplicate]
Let $f:\mathbb C \to \mathbb C$ be a function such that $(f(z))^2$ and $(f(z))^3$ are entire then is $f$ entire ?
I can conclude $f$ is entire if given $f$ is continuous ; but without continuity ...
user228168