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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7
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1answer
99 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 ...
6
votes
0answers
40 views

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 ...
5
votes
0answers
79 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
votes
0answers
243 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
0answers
426 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
44 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
233 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
votes
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 ...
2
votes
0answers
92 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
185 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
134 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
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 $$ ...
1
vote
0answers
17 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) ...
1
vote
0answers
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 ...
1
vote
0answers
23 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
17 views

Analytic semiconjugacy

Consider the following commutative diagram (semi-conjugacy): $$ X\;\; \stackrel{f}{\longrightarrow} \;\;X $$ $${\pi}\downarrow \;\;\;\;\;\; \;\;\;\;\downarrow {\pi}$$ ...
1
vote
0answers
26 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
vote
0answers
90 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
vote
0answers
25 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
vote
0answers
96 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
108 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 ...
1
vote
0answers
66 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
vote
0answers
21 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
vote
0answers
173 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 ...
1
vote
0answers
46 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
64 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
vote
0answers
95 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
13 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 ...
0
votes
0answers
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 ...
0
votes
0answers
40 views

entire function in complex analysis

$$ y = \left\{ \begin{array}{ll} \dfrac{\cos z}{z^2-\left( \dfrac{\pi}{2} \right)^2}, & z \ne \pm \dfrac{\pi}{2}\\ -\dfrac{1}{\pi}, & z = \pm \dfrac{\pi}{2}\\ \end{array} \right. $$ $$ ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 $f(z)=\frac{z^2+2}{z^2}$?
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
37 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) = ...
0
votes
0answers
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, ...
0
votes
0answers
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 ...
0
votes
0answers
32 views

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 ...
0
votes
0answers
42 views

Liouville's theorem for functions not bounded on an isolated set

On Liouville's theorem. Prove that if $f(z)$ is defined, analytic, bounded in the entire complex plane, except for an isolated set of points, then $f$ must be constant. What happens if said set has a ...
0
votes
0answers
19 views

Analytic Function, taking value 0 on an open ball

Let $f: D \rightarrow \mathbb{C}$ be analytic on a domain $D$. Suppose that $f(z)=0$ for all $z \in B(z_0, r) \subset D$ for some $z_0 \in D$ and $r>0$. Must $f \equiv 0$ on $D$?
0
votes
0answers
26 views

flabbiness of hyperfunctions

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...
0
votes
0answers
33 views

$f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
0
votes
0answers
15 views

Milne-thompson for polar form

How can we readily apply MilneThompson method to find the conjugate function if the complex variable is given in polar form? If a complex variable is given as $u(x,y) + i v(x,y)$ then its $f'(Z) ...
0
votes
0answers
21 views

Is $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ a general analytic function?

I have an expression $f(z)=\int_{-\infty}^{\infty}c(k)e^{-k^2/2}e^{ikz}dk$ where $c(k)\in\boldsymbol{C}$ and $k\in\boldsymbol{R}$. $f(z)$ is an analytic function, since it contains only non-negative ...
0
votes
0answers
42 views

Upper bound for integral where the integrand is not bounded on [r,1].

I'm reading an article of S. Rohde (On an Estimate of Makarov in Conformal Mapping, 1988) and there is a lemma which says that for univalent function it is possible to find a subset of $\partial D$ ...
0
votes
0answers
64 views

Determine whether this class of holomorphic functions is starlike

According to Singh and Singh [1], the class of holomorphic functions $f$ in the unit disk such that $f(0)=f'(0)-1=0$ and $\mathrm{Re}\{f'(z)+zf''(z)\}>0$ is a subclass of starlike function. My ...
0
votes
0answers
49 views

Finding Analytic Functions that Satisfy Certain Conditions

How would one go about finding: All analytic f such that $|f''(x)|>|e^{7z}|$? All analytic functions such that $f(z) = z + f(z^2)$. All linear functions that map from the punctured unit disk to ...
0
votes
0answers
54 views

Internal point transformed in an external one?

Let $f \colon \Omega \to \mathbb{C} $ be an analytic function over a connected open subset $\Omega$ of $\mathbb{C}$ and let $\gamma$ a rectifiable closed curve in $\Omega$. If $a$ is a point which is ...
-2
votes
0answers
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 $D$.