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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13
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169 views

Is this function nowhere analytic?

One usually sees $f(x):=\exp\frac{-1}{x^2}$ as an example of a $C^\infty$ function that is not analytic, having one point of non-analyticity (the point $0$). The Fabius function is a canonical ...
8
votes
0answers
117 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 ...
6
votes
0answers
204 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
votes
0answers
59 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. $f$ ...
5
votes
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185 views

Laurent expansion of $\frac{1}{\sin z}$

Question is a fully solved exercise in Gamelin's complex analysis. Exercise : Consider the Laurent series expansion for $\frac{1}{\sin z}$ that converges on the circle $\{|z|=4\}$. Find the ...
5
votes
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69 views

Finding an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$

I encountered the following problem in the lecture note in my complex analysis class: Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$. ...
5
votes
0answers
87 views

$f(x+1)=f(x)+f(\alpha\cdot x)$

I try to find an analytic increasing function $f_\alpha$ ($0\le\alpha\le1$) from $\mathbb R$ (or $\mathbb R^+$) to $\mathbb R$ such that for all $x$ $$f_\alpha(x+1)=f_\alpha(x)+f_\alpha(\alpha\cdot ...
5
votes
0answers
504 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 ...
4
votes
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159 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
votes
0answers
122 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
votes
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472 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
votes
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447 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
26 views

Does a analytic joint distribution necessarily have continuous marginals?

Although the question actually popped up in a course about evolution equations, it seemed most natural to ask this in the context of joint distributions. Namely: Suppose $X$ and $Y$ are two ...
3
votes
0answers
54 views

Show that $f(z) = \ln r + i \varphi$ is differentiable in a neighborhood of $z_{0}$

I am faced with the following problem: Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood ...
3
votes
0answers
55 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
3
votes
0answers
51 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
votes
0answers
164 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 ...
3
votes
0answers
95 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
votes
0answers
383 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
votes
0answers
70 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
votes
0answers
489 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
votes
0answers
36 views

$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
2
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46 views

A single analytic function that can approximate all others

The problem in it's entirety is this: Given some simply connected domain $U$ then show that there is a function $g\in\mathcal{O}(\mathbb{C})$ such that for any given $f$ there exists a sequence ...
2
votes
0answers
35 views

Verification of example to show surjective maps of sheaves need not surject onto sections in all open sets

As an exercise in understanding the notion of surjectivity in the category of sheaves, I came up with this example, slightly modifying the standard ones given in my textbooks. I feel like this one is ...
2
votes
0answers
93 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
2
votes
0answers
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
votes
0answers
36 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
votes
0answers
36 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
votes
0answers
244 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
votes
0answers
82 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
votes
0answers
32 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
votes
0answers
75 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
votes
0answers
76 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
votes
0answers
433 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 ...
2
votes
0answers
129 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
votes
0answers
225 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 ...
2
votes
0answers
90 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 ...
1
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0answers
26 views

Interpolating discrete data with completely monotone analytic functions

Suppose we have a positive integer $n$ and a finite list of real numbers $\{a_1,\,a_2,\,\dots,\,a_n\}$. We want to find a real-analytic function $f:[1,n]\to\mathbb R$ such that $f(m)=a_m$ for all ...
1
vote
0answers
37 views

function analytic in the entire complex plane is constant

Let the function $f$ be analytic in the entire complex plane and suppose that $\frac{f(z)}{z}\rightarrow 0$ as $|z|\rightarrow \infty$. Prove that $f$ is a constant. As $\frac{f(z)}{z}\rightarrow 0$ ...
1
vote
0answers
30 views

How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
1
vote
0answers
48 views

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges

Suppose $g$ is a continuous function such that the integral $\int_0^\infty|g(t)|dt$ converges. Is $$F(z)=\int_0^\infty g(t)\sin(zt)dt$$ analytic? And if so, in what region? My attempt: ...
1
vote
0answers
79 views

Simply connected domains and complex logarithms

While studying Complex Analysis from my professor's notes I came across the following theorem. A demain $D$ in the complex plane is simply connected if and only if any analytic function $f(z)$ on ...
1
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0answers
19 views

Composition of analytic functions is analytic in Manifolds

My problem is in analytic manifolds.According to Cohn's book a function $f$ in a manifold $M$ is analytic at $p \in M$ if it can be expressed as a power series of $\sigma(p)=(x_{0})$. That means ...
1
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0answers
28 views

Multisection of a Power Series Proof

Suppose that $$H_{N,k}(x)=\frac{x^ke^{\frac{-x}{N}}}{N^{k-1}k!\sum_{n=0}^{N-1}{w_N^{-nk}e^{\frac{w_N^nx}{N}}}}=\sum_{n=0}^\infty{A_n\frac{x^n}{n!}}$$ where $k\lt N, w_N=e^{\frac{2i\pi}{N}}$, and ...
1
vote
0answers
40 views

Is this result for entire functions with zeroes only at the origin… more basic than the Hadamard canonical product representation?

I just worked on a problem and was able to solve it pretty easily, using Hadamard's product representation. But I wonder whether the solution that I compared my work to doesn't actually use the ...
1
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0answers
16 views

Can an analytic function defined on a maximal torus be extended analytically to all the Lie group?

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?
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39 views

Show that an analytic function defined on unit ball with these properties does not exist

Show that there does not exist an analytic function $F : B(0,1) \to \mathbb{C}-\mathbb{Q} $ with $F(0)=i$ and $F(1/2)=-i$. I have already used the open mapping theorem to show that if we assume ...
1
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0answers
60 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
1
vote
0answers
29 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
1
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0answers
32 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 ...