Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
Questions tagged [cv.complex-variables]
3157 questions
75
votes
3 answers
Does a power series converging everywhere on its circle of convergence define a continuous function?
Consider a complex power series $\sum a_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a_n w^n $ converges .
We thus obtain a complex-valued function $f$…
Georges Elencwajg
- 46,833
66
votes
1 answer
Behaviour of power series on their circle of convergence
I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the unit circle $C$ can be realised as
$$
S:=\{x \in…
Piotr
- 663
45
votes
5 answers
Liouville's theorem with your bare hands
Liouville's theorem from complex analysis states that a holomorphic function $f(z)$ on the plane that is bounded in magnitude is constant. The usual proof uses the Cauchy integral formula. But this has always struck me as indirect and…
Jonah Sinick
- 6,942
42
votes
7 answers
Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?
The title pretty much says it all. I am revisiting complex analysis for the first time since I "learned" some as an undergraduate. I am trying to wrap my head around why it should be the case that a function which is differentiable once should be…
Steven Gubkin
- 11,945
37
votes
1 answer
Circles and rational functions
Suppose that $\gamma$ is a Jordan analytic curve on the Riemann sphere,
and there exist two rational functions $f$ and $g$ such that
$f$ maps $\gamma$ into a circle, and $g$ maps a circle into $\gamma$.
(All rational functions considered are of…
Alexandre Eremenko
- 88,753
36
votes
7 answers
Pathology in Complex Analysis
Complex analysis is the good twin and real analysis the evil one:
beautiful formulas and elegant theorems seem to blossom spontaneously
in the complex domain, while toil and pathology rule the reals. ~
Charles Pugh
People often like to talk…
Qi Zhu
- 425
32
votes
7 answers
Interpreting the Famous Five equation
$$e^{\pi i} + 1 = 0$$
I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us?
Best that I can figure out is that it just emphasizes that the various definitions…
Sunil Nanda
- 425
29
votes
4 answers
Entire function bounded at every line
I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.
FisiaiLusia
- 459
28
votes
1 answer
A holomorphic function sending integers (and only integers) to $\{0,1,2,3\}$
Does there exist a function $f$, holomorphic on the whole complex plane $\mathbb{C}$, such that $f\left(\mathbb{Z}\right)=\{0,1,2,3\}$ and
$\forall z\in\mathbb{C}\ (f(z)\in\{0,1,2,3\}\Rightarrow z\in\mathbb{Z})$?
If yes, is it possible to have an…
user1950
- 393
28
votes
1 answer
Are entire functions “essentially” determined by their maximum modulus function?
(Note: This has been asked on Math SE, but without an answer after almost two years and one offered bounty.)
For an entire function $f$ let $M(r,f)=\max_{|z|=r}|f(z)|$ be its maximum modulus function. $M(r, f)$ does not change
if $f$ is replaced by…
Martin R
- 490
28
votes
5 answers
Continuous + holomorphic on a dense open => holomorphic?
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the…
André Henriques
- 42,480
28
votes
2 answers
Restriction of a complex polynomial to the unit circle
I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).
Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write…
senti_today
- 1,284
23
votes
0 answers
Is analytic capacity inner regular?
For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by
$$\gamma(K) := \sup |f'(\infty)|$$
where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ in the complement of $K$ :
$f \in…
Malik Younsi
- 1,942
23
votes
4 answers
Lower bounding $|1+z+\cdots + z^{n-1}|$ when $z\approx 1$
I am trying to lower bound $|1+z+\cdots + z^{n-1}|$ when $z$ is a complex number close to $1$ (and $n$ is sufficiently large). My main concern occurs in the case $z = 1 + it$, where $t$ is small. In these cases, the terms of the geometric series…
David Altizio
- 347
23
votes
1 answer
On equation $f(z+1)-f(z)=f'(z)$
Original Problem
If $f$ is an entire function such that
$$ f(z+1)-f(z)=f'(z) $$
for all $z$.
Is there a non-trivial solution? ($f(z)=az+b$ is trivial)
And here is something uncertainty
If we use Fourier transform, how to define it to ensure any…
Lwins
- 1,531