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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6
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98 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
47 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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0answers
90 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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0answers
67 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
317 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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440 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
votes
0answers
49 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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33 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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0answers
49 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
301 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 ...
2
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28 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
35 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,\ ...
2
votes
0answers
37 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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0answers
53 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 $f$ ...
2
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0answers
49 views

Constructing an analytic continuation

I'm hoping someone could verify my answer to the following problem: Consider a function $f$ that is continuous for $Im(z) \geq 0$ and analytic for $Im(z) > 0$. Furthermore, assume that $f$ ...
2
votes
0answers
106 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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0answers
236 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 ...
2
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0answers
164 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 ...
1
vote
0answers
29 views

Zeta functions and their poles

consider a zeta function as follows $f(x):=\sum\limits_{m=1}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{\left(a\cdot m+n+\frac{1}{2}\right)^{2x}}$, for $a>0$ and $\Re(x)>1$. How can I construct ...
1
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0answers
16 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 $-1\leq z\leq 1$
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0answers
38 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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vote
0answers
58 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
35 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 ...
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0answers
37 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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0answers
37 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
42 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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0answers
70 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 ...
1
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0answers
31 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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50 views

Euler product for Riemann zeta and analytic continuation

the Euler product for the Riemann zeta $$ \zeta (s)= \prod _{p}\left( \sum_{n=0}^{\infty}p^{-ns}\right) $$ this is only valid for $ \Re(s) >1 $ however we could use the Borel transform so $$ ...
1
vote
0answers
29 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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0answers
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 ...
1
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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
vote
0answers
20 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
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0answers
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
110 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 $z^3+2i=0$ ...
1
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0answers
33 views

Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
1
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0answers
130 views

Schwarz Reflection Principle for function complex on the real line

The Schwarz reflection principle is usually proved for function real on the real (or a subset of) line. I wonder if the same principle/theorem works for general analytic functions on the real line? ...
1
vote
0answers
117 views

Applications of identity theorem to physics

Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the entire complex plane. So, just by knowing how the function behaves at a teenie-weenie open disc ...
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0answers
72 views

Infinite product of an analytic function on the right half plane

I want to know whether $G_t=\prod_{i=1}^\infty G(s)^i$ is defined and analytic on the right half plane for an analytic $G(s)$ on the right half plane. If so, is $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
1
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0answers
23 views

Analytic random function

Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function. I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic. What are the minimal conditions needed? ...
1
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0answers
198 views

Schwarz reflection principle and bounded derivatives

Suppose $f$ is a holomorphic function on $\Omega^+$ (an open subset of the upper complex plane) that extends continuously to $I$ (a subset of $\mathbb{R}$). Let $\Omega^-$ be the reflection of ...
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0answers
52 views

How to show that $f(x,y)$ are real analytical

Given a real function $f(x,y)$ on $R^2$, we know that (from wiki) it is called real analytic if it is locally given by a convergent power series. My question is that whether there has some principle ...
1
vote
0answers
66 views

Calculating germs for complete holomorphic function

I'm trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where ...
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0answers
97 views

Are nuclear Montel spaces projective?

The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
0
votes
0answers
33 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
0
votes
0answers
30 views

Proving that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$

Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this. Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now ...
0
votes
0answers
29 views

Approximating an L2 function by analytic functions

Spose I have a $ h \in L^2(U) $ where $ U \subset \mathbb{R}^3 $ is open and bounded. Is it possible to approximate this by analytic functions? If so, spose now we take $ U = \mathbb{R}^3 $. Is this ...
0
votes
0answers
29 views

Real analytic function in one point and zeros

Let $f:I\to\mathbb{R}$ be a continuous function and $t_0\in I$ such that $f$ is analytic at $t_0$ and $f(t_0)=0$. Is it true that there is an entire neighbourhood of $t_0$ in which $f$ has no other ...
0
votes
0answers
42 views

Sufficient Conditions for Analyticity: C-R equations and?

My understanding of the subject is: (1) One of the necessary conditions for analyticity of f(x,y)=u(x,y)+iv(x,y) is satisfaction of the Cauchy-Riemann equations (in x-y, polar, or z-z* form). (2) The ...
0
votes
0answers
20 views

Complex Plane - Analytic Function

I am trying to understand the definition of an analytic function and how to solve for it's domain. I understand that for $f(z) = {1\over z}$ the function is analytic on the complex plane except for 0. ...