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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1answer
22 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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1answer
97 views

$\forall x \,\exists k$ 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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1answer
37 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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0answers
38 views

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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44 views

Complex integration with complex integrands [on hold]

How to solve $$ \int_0^{1+i}(x-y+ix^2)dz$$
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34 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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32 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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57 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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134 views

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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1answer
32 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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28 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 ...
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52 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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23 views

Analytic function on bounded region

Let $f$ be analytic in a bounded domain $D$. $f(z)=f(2z)$ for every $z\in D$.Then show that $f(z)=0$ in $D$.
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2answers
50 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
31 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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210 views

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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34 views

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
33 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
25 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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36 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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41 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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23 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
20 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
17 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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1answer
52 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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20 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.
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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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1answer
19 views

Finding a Real valued Function to Create Holomorphism

I am asked whether it is possible to find a real function $v$ such that $$x^3+y^3+iv$$ is holomorphic. Should I basically be working backwards from the Cauchy-Riemann equations? That makes logical ...
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1answer
39 views

Integrating the function Im(z) on a variety of contours.

I've been asked to evaluate $\int_C Im(z) dz$ for a variety of contours, which I've had no issue in doing. For the sake of clarity, these contours included the upper and lower halves of the circle ...
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46 views

maximum modulus principle analytic function

I am trying to show that: Let $f$ be analytic on a given closed unit disc $D$ then prove that for every $k\in\mathbb N$ there is $w\in Bd(D)$ such that $|f(z)-w^{-k}|≥1.$ where z is in the unit disc ...
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42 views

Tthe inverse of a Mellin transform of a polynomial…

Let $\mathcal{M}$ be the symbol of the Mellin transform as define in http://en.wikipedia.org/wiki/Mellin_transform In a calculus, I finally end up with $$\mathcal{M^{-1}(f)}=\mathcal{P}$$ where ...
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1answer
67 views

Convergence Radius => Nonanalytic

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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48 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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36 views

Solution of Recurrence Relation for 1/2-integers

Suppose one wants to solve a recurrence relation of the form $$ c(m+1) - c(m)/f(m) = -g(m) $$ for $c(m)$. The general solution can be given by $$ c(m) = ...
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1answer
63 views

How to prove that $a$ is unique

Assume that $f : ℂ→ℂ$ is a non-constant non polynomial and entire function and there exist $a∈ℂ$ such that the fiber $f⁻¹(a)$ is finite. My question is: How to prove that $a$ is unique.
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24 views

Suppose f and g are analytic on a domain G

Suppose f and g are analytic on a domain G. If f and g are non-constant, then for any b in G, there exists a punctured disk D'(b,R) of radius R>0 such that f(z)g(z) is not equal to g(z)-f(z) + 1 for ...
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1answer
36 views

Relation between continuity of $f$ and analyticity of $f(z)^8$

If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
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144 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
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28 views

Analytic Map from $B(0,1)$ to $B(0,1)$

Is the analytic map from $B(0,1)$ to $B(0,1)$ such that $f(0)=1/2$ and $f'(0)=3/4$ unique?
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32 views

Maximum modulus principle, is it true?

Suppose f is analytic in an open set containing the open disk D(2+3i, 7) and its boundary circle C(2+3i, 7) such that |f(z) + 7i + 24|<25 for all z in C(2+3i,7). Then f has no zeroes inside D(2+3i, ...
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1answer
46 views

If composition with a linear functional is continuous, is the function continuous?

If $G$ is an open subset of $\mathbb{C}$ and $f:G \to X$ is a function such that for each $x^*$ in $X^*$, $x^*\circ f:G\to\mathbb{C}$ is analytic, then f is analytic. Will the statement still hold ...
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38 views

Analytic Continuation of a Function Containing a Square Root to a Second Riemann Sheet

Consider the function $f(z) = g_1(z) + \sqrt{z} \, g_2(z)$, where $g_1(z)$ and $g_2(z)$ are entire functions, and we take the principal branch of the square root. $f$ is analytic on $\mathbb{C} / \{z ...
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Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
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74 views

The Identity Theorem for real analytic functions

What is the condition for two real analytic functions to be identically equal? We know that there is a nice condition (Identity Theorem) for holomorphic function to check if they are the same. What is ...
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22 views

Showing a function is not analytic

Let $g(x)=e^{-1/x^2}$ for $x\neq0$, and $g(0)=0$. I've shown that $g^{(n)}(0)=0$ and thus that it is infinitely differentiable about 0, but now I must show that it cannot be expressed as a power ...
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45 views

Proofs for $g(x)=e^{-1/x^2}$ when $x\neq0$, and $g(x)=0$ when $x=0$

Sorry for the non-descriptive title - the question is a bit long. I have $g(x)$ as in the title, and we proved previously that $g'(0)=0$ using L'Hôpital's rule. Now I must show by induction that ...