**74**

votes

**41**answers

21k views

### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...

**14**

votes

**4**answers

2k views

### Conformal maps in higher dimensions

In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( $\neq \mathbb{R}^{2}$) can be mapped bijectively to the unit disk with a function that preserves angles between ...

**2**

votes

**2**answers

428 views

### $ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear

I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is ...

**5**

votes

**3**answers

372 views

### Visualizing functions with a number of independant variables

I need to graph real valued functions ( for exposition and analysis)
the issue is the independent variables are more so that the conventional graphing method cant be used and further i don't want to ...

**3**

votes

**1**answer

196 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

**0**

votes

**0**answers

106 views

### Fractional Derivative of A specific function

I've tried and tried to do this problem myself, but I've hit some snags on the way.
I'm trying to take the fractional derivative of:
$f(x)=1+n^{-x}$ where n is an integer and $n\geq2$ and $x>1$.
...

**93**

votes

**1**answer

8k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**45**

votes

**3**answers

2k views

### 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 ...

**24**

votes

**5**answers

2k views

### 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 ...

**16**

votes

**10**answers

4k views

### Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...

**26**

votes

**1**answer

2k views

### 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 ...

**16**

votes

**2**answers

820 views

### Maximum of a function of one variable

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...

**25**

votes

**1**answer

5k views

### What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi,
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...

**9**

votes

**2**answers

1k views

### Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations ...

**5**

votes

**3**answers

769 views

### When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...

**12**

votes

**5**answers

1k views

### 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 ...

**8**

votes

**0**answers

355 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**6**

votes

**2**answers

691 views

### Characterize where the Dirichlet Problem for the Laplacian is always solvable

Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...

**3**

votes

**4**answers

1k views

### Minimizing the modulus of a polynomial around a circle

I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...

**2**

votes

**0**answers

373 views

### Relation between entire function of exponential type and exponential polynomials

Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...

**12**

votes

**1**answer

425 views

### How can one “see” the Hopf fibration in the space of lattices in the plane?

This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006.
The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...

**9**

votes

**6**answers

576 views

### Functions holomorphic on a region minus a Cantor set

Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is ...

**4**

votes

**1**answer

206 views

### Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...

**4**

votes

**1**answer

675 views

### Real integral which cannot be evaluated without complex analysis?

One motivation for studying contour integrals in complex analysis is that they can be used to give elegant evaluations of certain real integrals, which tend to have more direct physical and ...

**3**

votes

**1**answer

447 views

### Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)

What are modular forms or cusps forms, resp. ?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The ...

**3**

votes

**4**answers

2k views

### Analytic implicit function theorem

I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic ...

**2**

votes

**0**answers

155 views

### What can be said about a function given its asymptotic expansion?

This is probably not a research level question but I honestly don't know how/where to look for techniques to reconstruct a function from its asymptotic expansion.
The expansion I want to know about ...

**5**

votes

**1**answer

401 views

### Hurwitz's automorphisms theorem with deformations

Hurwitz's automorphisms theorem bounds the order of the automorphism group of a negatively curved Riemann in terms of the genus.
Now suppose a finite group $G$ acts faithfully on a Riemann surface ...

**3**

votes

**1**answer

286 views

### Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...

**3**

votes

**0**answers

235 views

### Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...

**3**

votes

**0**answers

120 views

### When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?

If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when is ...

**0**

votes

**0**answers

63 views

### Finding the Fractional Derivative of This Function [duplicate]

I've been trying to find an answer to this question, and it seems as though the question has gone unanswered. The question regards the derivative of $f(x)=1+n^{-x}$ where $n$ is a natural number. Is ...