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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17 views

Question about constructing complex entire function

How to construct an entire function for infinitely many prescribed values? i.e. I hope to find an entire function $f$ given $f(z_k) = w_k$ ($w_k$ might not be zero) for a sequence of $\{z_k\}$ with ...
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18 views

Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ and find $f(0)$ and $f'(0)$

So I have found this problem in the last exam paper, but I have never done anything like this in class. The professor says that the resit will be very similar to the last 2 papers, so I would really ...
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1answer
17 views

Moment-determinacy in multivariate case

Let $X$ be a random vector with probability density $p$. In the scalar case I have learned that if the characteristic function of $X$ is real analytic, then all moments exist and $p$ is determined ...
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2answers
41 views

Complex function defined by contour integral along a smoothly varying path

Let $D$ be a domain in the complex plane. Consider the function $F: D\to \mathbb{C}$, defined by $$ F\left( z \right) = \int_{\mathscr{C}\left( z \right)} {f\left( {z,t} \right)dt} . $$ Suppose that ...
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36 views

Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
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88 views

A real analytic function that takes each value in $\mathbb{R}$ three times

I was inspired by this question: it is quite easy to prove that for any positive odd number $2m+1$ there exists a function $f\in C^{\infty}(\mathbb{R})$ such that ...
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1answer
33 views

Upper bound for modulus of a function

Let $f(t,x)$ be a bounded and continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_x$. Moreover, assume that for each fixed $t$, ...
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35 views

Does analytic at a point implies differentiable at that point?

A function $f:A \subset \Bbb C \to \Bbb C$ is said to be analytic if it is locally given by a convergent power series i.e. every $z_o \in A$ has a neighbourhood contained in $A$ such that there exists ...
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28 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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23 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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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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31 views

Analytic Function In The Complex Plane Which Always Gives Real Values

Let $f$ be a non-constant, complex-valued function which is defined and analytic for all $z$ in the complex plane. Also, $f$ has the additional property that it is always real. To me, such a function ...
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291 views

What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
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2answers
37 views

Proof that Radius of Convergence Extend to Nearest Singularity

Can someone provide a proof for the fact that the radius of convergence of the power series of an analytic function is the distance to the nearest singularity? I've read the identity theorem, but I ...
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30 views

Analytic Extension: Imaginary Stripe

I was always wondering the following: Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends ...
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41 views

Is there an analytic function defined on $\Bbb C$ except for Gaussian integers where it has poles of order 1 and residue 1?

I need a function defined for all complex variables $z$, except for at all the Gaussian integers, where it has poles of order 1 and residue 1. The function has to be complex-analytic. Can anyone ...
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31 views

Covering up discontinuities to create analyticity

The floor function, $\lfloor x \rfloor$ , has a "jump" at the integers where its derivative ceases to exist. Everywhere else, its derivative is zero. Now, I wish to multiply the floor function by ...
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49 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 ...
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39 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. $f$ ...
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1answer
35 views

Uniformly analytic functions

Consider the following definition: Let $\Omega$ be an open set of $\mathbb{R}_x^n$, $x = (x_1, ..., x_n)$. A $\mathcal{C}^{\infty}$-function $\varphi(x)$ on $\Omega$ is said to be uniformly analytic ...
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Radius of convergence continuous?

Let $ f: [0,1] \rightarrow \mathbb{R} $ be analytic. Let $ r_f(x) $ be the radius of convergence of $ f $ at $ x $. Is $ r_x(f) $ continuous? Alternatively, is there an $ r_{min} $ I can choose so ...
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33 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 $f$ ...
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$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show ...
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50 views

Boundary behaviour of holomorphic function on unit disk

Let $\mathbb{D}=\{z \in \mathbb{C} \ | \ |z|<1 \} $ be the open unit disk in the complex plane. I would like to see explicit examples of the following phenomena: a holomorphic function $f$ on ...
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Can we find a real $s$ such that $f(s)=w$ and $f'(s)≠0$?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
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40 views

Riemann removable singularity theorem for annuli

Let $\mathbb{D}^*=\{z \in \mathbb{C} \ | \ 0 < |z| < 1 \}$ denote the unit punctured disk in the complex plane. Riemann's theorem about removable singularities in particular implies the ...
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44 views

What is the difference between the terms smooth, analytical e continuous?

I saw the following (“roughly speaking”, like the author says) definition of a Lie group in ‘Group theory in Physics’, by Wu-Ki Tung: “Roughly speaking, a Lie group is an infinite group whose ...
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3answers
62 views

Does there exist an analytic function that satisfies these properties?

Does there exist an analytic function $f:\{z\in\mathbb{C}:0<|z|<1\}\to\mathbb{C}$ such that $\displaystyle\lim_{z\to0}[z^{-3}f^2(z)]=1$? I'm assuming that there is not such a function, so I've ...
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36 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
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36 views

Identity theorem for polynomials in several variables

Let us assume that we are given two polynomials $f,g$ with real coefficients in several variables, say $x_1, \ldots, x_n \in \mathbb{R}$. Further, assume that $f_{|X} \equiv g_{|X}$, with $X$ being ...
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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 ...
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2answers
56 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
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1answer
72 views

What is the notation for taking negative imaginary values for roots of negative numbers?

I have a formula which is analytic in its argument $x$. In it, there is a square root of a variable as in $\sqrt{x}$. Although meaningful results are obtained when positive roots are taken for for ...
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1answer
37 views

Analytic continuation of a real function

I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique. Now i am wondering if this is ...
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Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
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Is composition of analytic functions itself analytic?

Is composition of analytic functions itself analytic? Is there a proof that, say, $$f(x)=e^{\frac{x^2+1}{x^2-1}}$$ analytic?
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40 views

entire function in complex analysis

$$ y = \left\{ \begin{array}{ll} \dfrac{\cos z}{z^2-\left( \dfrac{\pi}{2} \right)^2}, & z \ne \pm \dfrac{\pi}{2}\\ -\dfrac{1}{\pi}, & z = \pm \dfrac{\pi}{2}\\ \end{array} \right. $$ $$ ...
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1answer
15 views

Extending a holomorphic function to a radial limit function for almost every angle

I've read in several places about the "well known theorem" which states that a holomorphic function on the (open) unit disk $D=\{z\in\mathbb{C}:\ |z|< 1\}$ can be extended to its boundary on almost ...
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1answer
34 views

Calculating an integral with a branch cut, using some “uniqueness property”

Consider a complex function $$\tilde{f}(z)=z\int_{M}^{\infty}ds' \frac{\rho(s')}{z-s'} \qquad (1)$$ , where $M>0$ and $$\rho(s')=\frac{1}{s'}\sqrt{1-M/s'}.$$ This function is analytic in the ...
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1answer
28 views

Finding the number of zeros of $f(z) = z^n$ if $|f(z)| < 1 $ for all $z$ with $|z|=1$.

Suppose $f: \overline{\mathbb{D}} \to \mathbb{C}$ is continuous, analytic in $\mathbb{D}$ and satisfies $|f(z)|<1$ for $|z|=1$. Find the number of solutions to the equation $f(z) = z^n$ where $n$ ...
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39 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 $$ ...
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86 views

An infinite compact set which allows no boundedness and analyticity

I need an example of an infinite compact set $K$ in $\mathbb {C}$ such that there does not exist any non-constant function which is both bounded and analytic on $\mathbb{C} - K$. First, any hints ...
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1answer
42 views

Integral Continuation $\Gamma(z)=\int_{0}^{1} e^{-t} t^{z-1} dt +\int_{1}^{\infty} e^{-t} t^{z-1}dt$

I am trying to obtain an analytical continuation for $\Gamma(z)$ into the region of the complex plane characterized by $\Re(z) \leq 0$ but am stuck. Starting from the integral definition of ...
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24 views

Show existence of an analytic which cannot be extended beyond the boundary

$G$ is an open strip $\{z:1<\text{Im } z<2\}$. Prove that there exists an analytic function $f(z)\in H(G)$ that does not extend analytically beyond any boundary point of $G$. Also determine ...
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1answer
24 views

To show a function is analytic

Let $G\subset\mathbb C$ be open and connected, and function $h$ is analytic on $G$. $\{f_n(z)\}$ is a sequence of analytic functions on $G$ for which $\lim_{n\rightarrow \infty}f_n(z)$ exists for any ...
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1answer
19 views

Termwise differentiation of sequence of rational functions when the uniform limit is analytic

Given a sequence $\{f_n(x)\}$ of rational functions which converges uniformly to the analytic function $\{g(x)\}$ on $[a, b]$ ($f_n(x)$ are defined on $[a, b]$ and hence are analytic), what can we say ...
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59 views

When is a Fourier series analytic?

By Fourier theory, every continuously differentiable function $f : S^1 \to \mathbf C$ admits a unique, uniformly convergent Fourier expansion $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...
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13 views

Argument principle of $f(z)=\frac{z^3+2}{z}$

let $x$ denote the unit circle $|z|=1$, described in the positive sense use theorem to determine the value of arg $f(z)$ when $f(z)=\frac{z^3+2}{z}$ and $f(z)=\frac{z^2+2}{z^2}$?
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21 views

existence of an analytic function in unit disk

Does there exists an analytic function $f$ in unit disk such that $f(-\frac{1}{2})=3$, $f(n^{-2})=5$ for $n\ge 2$. i am not able to solve any help will be appreciated.