A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties ...

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114 views

Why are numeric methods the only technique available to solving $\ln(x) = \sin(x)$? Is this $x$ transcendental?

I just read this question about finding the solution to the equation $\ln(x) = \sin(x)$. All the answers focus on using a numerical method to approximate the solution. This is interesting in its own ...
6
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0answers
51 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
6
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93 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
4
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64 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
4
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86 views

Riemann Zeta Function Analytic Continuation

I am struggling to understand how the analytic continuation of the Riemann Zeta function is derived to extend it to all complex values $z$ not equal to $1$, starting with the series which converges ...
4
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0answers
365 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
4
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441 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
3
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0answers
41 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
3
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72 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
3
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137 views

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
3
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46 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
3
votes
0answers
56 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
3
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0answers
312 views

Zeros of analytic function and limit points at boundary

Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose ...
3
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0answers
339 views

Question Relating with Open Mapping Theorem for Analytic Functions

This problem is taken from Section VIII.4 of Theodore Gamelin's Complex Analysis: Let $f(z)$ be an analytic function on the open unit disk $\mathbb{D}=\{|z|<1\}$. Suppose there is an annulus ...
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23 views

How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
2
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20 views

Application of Bernstein's theorem

There is a theorem due to Bernstein related to analytic functions : If $f : ]0,1[ \to \mathbb{R}$ is an absolutely monotonous function (that is a $\mathcal{C}^\infty$ function such that for ...
2
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0answers
17 views

What are some general strategies to build measure preserving real-analytic diffeomorphisms?

One could prove the following theorem in the smooth setting: Theorem Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be ...
2
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0answers
78 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...
2
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0answers
59 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
2
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0answers
30 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
2
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0answers
44 views

Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point

Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point. If i look at the limit of a more simple function of this form: f(z) = $\frac{z}{z^*}$ I would say that the limit does not exist, ...
2
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59 views

Non-Trivial Self-Inverse Analytic Function In The Complex Plane

Let $f$ be a complex-valued function which is analytic for all values of $z$ in the complex plane and which satisfies $$f(f(z))= z $$ i.e. it is a self-inverse function. The trivial solutions for $f$ ...
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112 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
2
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197 views

Prescribing zeroes, poles, principal parts and finitely many terms with positive exponents in Laurent series

I was given a problem to prove a theorem by Mittag-Leffler about prescribing the items in the title, using Weierstrass's theorem about prescribed zeroes and Mittag-Leffler's theorem about prescribed ...
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0answers
12 views

Computing the laplacian Green function by Fourier transform and analytic continuation

I know that the Green function for the laplacian operator in $d$ space dimensions $$\Delta \equiv \sum_{i=1}^d \frac{\partial^2}{\partial x^2_i}$$ is given by $$ \Delta^{-1}(x-x')=\begin{cases} ...
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0answers
23 views

How to apply Cauchy-Kowalevsky Theorem.

The Cauchy-Kowalevsky theorem is stated in my notes as: For the Cauchy problem: $$ \begin{cases} u_{y}=F(x,y,u,u_{x}) \\ u(x,0)=h(x) \end{cases} $$ If $h$ is analytic in a neighborhood of ...
1
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0answers
20 views

Schwarz Reflection Principle for the four quadrants in the plane and for two intersecting circles,

I'm looking at an old exam problem that shows a picture of what the function f does to the plane. On the upper right quadrant, there is a + sign, which indicates that f maps this quadrant one-to-one ...
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32 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
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18 views

Analytic structures on $S^1$|

I am currently studying Haefliger's paper "Homotopy and Integrablity". During the last chapter, he applies his theory of $\Gamma$-structures to analytic codimension $1$ foliations. Throughout the ...
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0answers
12 views

About smooth function whose reciprocal is also smooth

I know that for exponential functions $e^{at}$, or functions like $1/t^m$, t>0, both they and their reciprocals are smooth. Could you please give me more classes of smooth functions, or Analytic class ...
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0answers
42 views

Finding region where $f(z)=z^2\bar{z}$ is analytic.

How can I find a region where $f(z)=z^2\bar{z}$ is analytic ? I first let $z=x+iy$ ,then use Cauchy-Riemann equation and obtain $$u(x,y)=x^3+xy^2$$ $$v(x,y)=y^3+yx^2$$ $$u_x(x,y) = 3x^2 + y^2$$ ...
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0answers
44 views

Show that the function $f(z)$ is analytic

Question : If $\phi$ and $\psi$ ae function of $ x $ and $y$ satisfying laplace's equation . Show that $f(z) = s + it$ is analytic , where $$ s = \frac{\partial \phi}{\partial y} - \frac{\partial ...
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0answers
18 views

Is definite integral of such function multiplication analytic?

If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about $$H(s)=\int_{a}^b f(x) g(s,x) dx $$ Is $H(s)$ analytic ...
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0answers
56 views

Why is $1/z$ analytic at infinity?

I was given this proof: Let $w(z)=1/z$, so $w$ maps origin to inifinity and infinity to origin. Consider $f(z) = z$. It has no singularities in finite $z$-plane. So $f(w) = 1/w$ has a pole at the ...
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0answers
38 views

Single value function defined in the plane cut

I'm confused about multi/single value complex function. Can someone explain why the function $\sqrt{1-z^2}$ can be thought of as a single valued in the plane cut along $-1\leq z\leq 1$
1
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39 views

Zero of non analytic function

Let a function $L=L(z)$ be analytic, for $\mathrm{Re}\, z>0$, and be singular at $0$, however, $L(0)=c$ be finite. Let also $L'(0)$ be finite as well, however, $L'(0)\neq 0$. For example, $$ ...
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0answers
59 views

Number of connected components of $f^{-1} (U)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an analytical function (semialgebraic,polynomial if needed), $U$ be an open connected subset of $\mathbb{R}$. What can we say about the nuber of connected ...
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0answers
55 views

Solutions of complex equation for analytic function

Suppose $g(z)$ is an analytic function on the upper plane, in fact is constructed by Hilbert transform, \begin{equation} g(z) = g_R(x,y) + i g_I(x,y) \quad g_I = \mathcal{H}( g_R) + \text{real ...
1
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0answers
56 views

Complex analysis question, maximum principle application

Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty $ and $\alpha<1$ such that ...
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40 views

linear first order ODE with polynomial coeffficients

I'm wondering if anything can be said about the solution to a system of linear first order ODEs with polynomial coefficients, especially about analyticity of the solution. The equation is given as ...
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0answers
53 views

Preimage of the set of critical value of an analytic function between smooth manifolds

I have some problems with the following exercise, maybe due to alack of knowledge: Let $M$ be a connected smooth manifold and let $$ f \colon M \to N$$ be an analytic map. Denote by $C_f \subset M$ ...
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92 views

Zeros of the derivatives of a finite Blaschke product.

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known ...
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0answers
36 views

separate vs joint real analyticity

Let $$f(x,y) := xy\exp\left(-\frac{1}{x^2+y^2}\right),$$ if $(x,y)\neq (0,0)$ and $f(0,0):=0$. I read the claim that $f$ is (a) separately real analytic on $\mathbb{R}\times\mathbb{R}$ (i.e. for ...
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0answers
220 views

Uniqueness of Analytic Continuation

I wasn't very well introduced to Analytic Continuations, but from what I have seen, showing that the analytic continuation is unique is pretty simple. In Real Analysis, from what I can imagine, there ...
1
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0answers
37 views

Existence and uniqueness of an analytic function

I'm reviewing complex for the exam and just got stuck here. Let $g$ be an analytic function at $z=0$. We want to show there exists a unique analytic function $f$ such that (1) $f(0)=0$ (2) ...
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53 views

How to explain this result due to Pôlya

How to explain this result due to Pôlya: An entire function is determined uniquely by the inverse images, counting multiplicities of three distinct non omited values. I cannot understand how this ...
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0answers
25 views

Show that $f$ and $g$ are holomorphic in the set $D=({s=α+iβ∈ℂ: 0<α<1})$

Let us consider two complex functions $g,f$: $$g(α+iβ)=∑_{n=2}^{m}(-1)ⁿ⁻¹((n^{2α-1}-1)/n^{α})n^{iβ}$$ $$f(α+iβ)=(-1)^{m}(((m+1)^{2α-1}-1)/(m+1)^{α})(m+1)^{iβ}$$ My question is: Show that $f$ and ...
1
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0answers
21 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
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30 views

When is meromorphic continuation possible?

Suppose I have an expression of the form $$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region ...
1
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0answers
123 views

Determine where the function $f(z)=\operatorname{Log}(z^3+2i)$ is analytic.

I need to know if my intuition is correct here. Idea: I would use De'Moivre's formula to find all third roots of -2i and exclude one of these rays because these are the values that give $z^3+2i=0$ ...