Questions tagged [analyticity]
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 of functions over the complex numbers.
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questions
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Real Analytic continuation
For which values of $p,q\in[1,\infty)$ the following functions have a real analytic continuation to the whole real line.
1.$f:(0,\infty)\to \mathbb R$ where $f(t)=(1+t^p)^{\frac{q}{p}}$.
2.$g:(-\...
3
votes
1answer
23 views
Bounded number of zeros derivatives can have implies analyticity
I'm trying to prove that if $f\in C^{\infty}(]-1,1[,\mathbb{R})$ and there exists a $p \in \mathbb{N}$ so that for all $n\in\mathbb{N}$, $f^{(n)}$ has at most $p$ zeros in $]-1,1[$ then $f$ is ...
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1answer
64 views
Holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$
Does there exist a holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$ where $ D= \{ z \in \mathbb{C} : |z|<1\}$.
I cannot use any of Schwarz lemma or ...
1
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0answers
27 views
Use of the Identity Theorem?
I have the following question:
Let $s(y)$ and $t(y)$ be real differentiable functions on $y$ with $-\infty < y < \infty$ satisfying $s(0) = 1$ and $t(0) = 0$, with the property that the ...
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0answers
24 views
Why is this function defined by iteration analytic?
The function I'm dealing with is reached by solving iteratively the following integral equation ($x$ is real, and $ u $ is a real smooth function that goes to zero on infinity):
$$ \chi(x) = 1-\int_{...
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votes
1answer
29 views
Am I applying MAX Modulus principle correctly here?
$(1)$ Find all possible entire functions s. t. $$|f(z)|\leqslant2|z|+1.$$
$(2)$ Prove you have found all such functions.
According to my understanding of $MAX$ modulus principle, $|f(z)|$ doesn' t ...
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3answers
56 views
Analytic gradient $\nabla f$ implies analytic $f$
I'm wondering how to prove that a real function $f : \mathbb{R}^d \to \mathbb{R}$ with analytic gradient $\nabla f$ (equivalently, analytic Fréchet derivative $Df$) must also be analytic. We can ...
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0answers
28 views
Meaning of “analytic function” in the context of Banach algebras
In going through Folland's Abstract Harmonic Analysis, I came on the following. Let $\mathcal{A}$ be a unital Banach algebra with unit $e$, and define $$\sigma(x) = \{ \lambda \in \mathbb{C} : \lambda ...
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0answers
14 views
Cauchy-Riemann equations and definition of analytic function at point.
investigate the analicity of $f(z)=r^2cos^2\theta+ir^2sin^2\theta$
I formed the Cauchy Riemann equation and it turns out , the C-R eqs hold in two cases below:
1-$r=0$ ( origin )
2-$sin\theta=cos\...
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1answer
30 views
Theorem 1.1 Part 2 Lang's Complex Analysis proof explanation
Page 294, Part 2 chapter 1 in Langs "Complex Analysis."
Let $U^+$ be an open set in the upper half plane, and suppose $\delta U^+$ contains an interval $I$ of real numbers. Let $U^-$ be the ...
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0answers
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Prove that an analytic function bounded by a certain exponential is bounded. [duplicate]
Let $D=\lbrace z \in \mathbb{C} : Re(z)>0, \frac{-\pi}{2} < Im(z) < \frac{\pi}{2} \rbrace$ and $f$ a function analytic on $D$. Suppose $|f| \leq 1$ on the boundary of D and that there exists ...
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0answers
19 views
Given analytic function $f$, why is $f: (w,z) \mapsto (f(z)-f(w))(z-w)^{-1}$ continuous? [duplicate]
We are given an analytic function $f$ on an open subset $G\subseteq \mathbb{C}$. I want to show the function $\phi$ on $G \times G$ given by
$$(z,w) \mapsto \frac{f(z)-f(w)}{z-w}$$ if $z\neq w$ and $$...
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1answer
58 views
Apply Cauchy-Riemann equation on $\sqrt{z}$ to show it is analytic
I want to show that the function $f(z) = \sqrt{z}$ is analytic on $D = \{z\in\mathbb{C}:Re(z)>0\}$ by the Cauchy-Riemann equation.
But here is the thing. I fail to rewrite $f(z)$ into $u(x,y) + iv(...
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1answer
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Zero set of a non constant analytic function. [closed]
Is there any example of a non constant analytic function on { z : |z|<1} , which have infinite zeros in that domain?
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1answer
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$z=0$ is a Pole of order $2$
I was trying to find order of $z=0$ for $$f(z)=\frac{\cos z}{z\sin z}$$
We have $$g(z)=z\sin z$$
Evidently $g(0)=0$
Also $$g'(z)=z\cos z+\sin z$$
So $g'(0)=0$
Now $$g''(0)=\cos z-z\sin z+\cos z$$
...
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0answers
23 views
Extension of Fredholm analytic theorem
Is there any extension result of the classical Fredholm analytic theorem when the domain of the operator is also analytically-dependent of the parameter $\lambda$ ?
1
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1answer
28 views
Analytic function problem
Let $I \subseteq \Bbb R$ be non-empty open interval. Let $f : I \rightarrow \Bbb R$ be a real analytic function. Let $y_0 \in I$ be a point such that $f^{\prime}(y_0) \neq 0$
(a) Show that there is ...
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0answers
23 views
Function which is analytic except for countably many vertical lines and continuous also there
Suppose I have a function $f$ of which we know that it is analytic in the set $\mathbb{C}\setminus S$, where $$S := \bigcup_{n=-\infty}^\infty S_n := \bigcup_{n=-\infty}^\infty\{ z \in \mathbb{C} : ...
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1answer
49 views
If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$. [closed]
If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$.
I could not find any confirmation regarding this solution of $a$ and $b$.
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0answers
44 views
find all functions f, for which g is analytic
let f(z)=u+iv be analytic in open disk $D=\{z\in C s.t |Z|<1\}$.
assume $u\neq v$ for all (x,y) in D.
Find all such f, for which the funstion $g(z)=u^2+iv^2$ in analytic in D.
My answer is: g is ...
5
votes
1answer
164 views
Are two analytic functions equal if they are equal on the boundary of an open disk?
Let $D$ be the open disk in $\mathbb{C}$ with origin $0$ and radius $1$.
Let $f,g: \overline{D} \to \mathbb{C}$ be continuous functions such that $f$ and $g$ are analytic on $D$ and such that $f=g$ ...
1
vote
2answers
28 views
Products of power series
consider the identity
$$\frac{e^{-x}}{1-x}=\sum_{n=0}^{\infty}c_nx^n$$
Show that for each $n\ge0$
$$\sum_{k=0}^{n}\frac{c_k}{(n-k)!}=1$$
My trial :
By cauchy product,
$$c_k=\sum_{i=0}^{k}\frac{(-...
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0answers
29 views
Is this proof of $(A(z)f,\varphi)$ analytic implies $A$ analytic correct?
Let $(A(z)f,\varphi)$ be analytic for any $f$, $\varphi$ ($\varphi$ in dual) where $A(z)$ gives a function from Banach space to Banach space. The variable z belongs in open subset of $\mathbb C$. I ...
0
votes
1answer
60 views
Analyticity implies continuity of first derivative
I will cite the relevant definitions and theorems I am working with from Wunsch's Complex variables with applications.
Def. (Analyticity) A function $f(z)$ is analytic at $z_0$ if $f'(z_0)$ exists ...
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0answers
57 views
prove you have found all such functions
find all possible entire functions f with the property that $|f(z)|\le2|z|+1$ for all $z\in C$. Prove that you have found all such functions.
First of all I am self studying complex analysis so sorry ...
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1answer
38 views
Solving A System Of PDE by Matlab
I have a system of PDE which I would like to solve it by Matlab(numerically or analytically). How can I do this? Are there any known analytical approaches to problems of this kind?
$$\frac{\partial ...
6
votes
1answer
140 views
Why is the sum of two algebraic functions algebraic?
Let $U\subset\mathbb{C}^n$ be a domain. A holomorphic function $f:U\to \mathbb{C}$ is called $\textbf{algebraic}$ if there exists a polynomial $p(x,y)$ in the variables of $U\times \mathbb{C}$ such ...
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0answers
5 views
does the scaling function of the coiflet wavelet have an analytical expression?
so I'm simply wondering if there exists an analytical expression for the scaling function associated with the Coiflet wavelet (specifically in the case S = 6).
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0answers
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Analytic continuation of a Gaussian expectation w.r.t covariance matrix
Let $(N_1,N_2,N_3)$ a Gaussian vector with covariance matrix :
$$A = Cov(N_1,N_2,N_3) = \begin{pmatrix}
1 &0&b\\
0&1&c\\
b&c&1
\end{pmatrix}$$
Let $f$ be some function s.t. $\...
2
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2answers
69 views
Proving that the limit of a function doesn't exist at an essential singularity
Assume $f$ is analytic on a deleted neighborhood $B'(0;a)$. Prove that
the limit of the function as $z$ approaches $0$ exists (possibly
infinite) if, and only if there exists an integer $n$ and a ...
1
vote
1answer
22 views
Verifying if a function is analytic
I'm trying to solve the third exercise from page 126 of Conway's complex variables textbook. It says:
Let $f$ be analytic in $\bar{B}(0,R)$, with $f(0)=0$, $f'(0)\neq 0$ and $f(z)\neq 0$ for $0<|z|...
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2answers
147 views
Proof that $H(z)=\int_0^\infty h(z,t)\,dt$ is analytic?
Let $h(t,z)$ be a continuous complex-valued function defined for $0\leq t<\infty$ and $z\in D\subset\mathbb C$, where $D$ is a domain. Suppose that for each fixed $t$, $h(t,z)$ is an analytic ...
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0answers
34 views
Conformal map from a 7-sided polyhedron to a square pyramid.
I have a right-angled square pyramid, $A$, whose height and base-length is $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
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0answers
35 views
Conformal map from a quadrilateral to a triangle
I have a right-angled isosceles triangle, $A$, whose length and height are $l$. Now supposed I form a trapezium, $B$, by gluing a square, whose side-length is $l$ as well, to the vertical edge of $A$ (...
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2answers
73 views
Complex Analysis 2.1.2 - 1 If g(w) and f(z) are analytic functions, show that g(f(z)) is also analytic. [closed]
Complex Analysis 2.1.2 - 1
If $g(w)$ and $f(z)$ are analytic functions, show that $g(f(z))$ is also analytic.
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0answers
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Doubt regarding proving analyticity of a power series
I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications"
In the chapter of analytic continuation in basic concepts authors mention that
...
1
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1answer
99 views
$x\mapsto \frac{1}{1+x^2}$ is analytic
Show that $f:\mathbb{R}\to\mathbb{R}, f(x)=\frac{1}{1+x^2}$ is analytic, i.e. that $\forall x_0 \in \mathbb{R}$, $f$ can be approximated in a neighbourhood of $x_0$ by a power series centred at $x_0$.
...
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$J(z)= \frac{1}{\pi} \int _0 ^\infty \frac{z}{\eta^2+z^2} \ln (\frac{1}{1-e^{-2\pi \eta}}) d \eta$ is analytic in Re $(z)>0$
In the Complex Analysis book of Ahlfors, the Stirling formula is written as follows:
Stirling's Formula. $\Gamma(z)= \sqrt {2 \pi} z^{z-1/2} e^{-z}e^{J(z)}$ (Re$ z>0$, where $J(z)= \frac{1}{\pi} \...
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0answers
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Does here exists an analytic function $f:\mathbb C → \mathbb C$ such that $f(0) = 1, f(4i) = i$…
Check whether the statement true or false?
There exists an analytic function $f:\mathbb C → \mathbb C$ such that $f(0) = 1, f(4i) = i$ and for all $z_j$ such that $1 < |z_j | < 3, j = 1, 2$, we ...
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0answers
33 views
$f$ analytic on $G\setminus \{a\}$ and bounded at $a$
Suppose $G$ is an open subset containing a point $a$; $f$ is analytic on $G\setminus \{a\}$ and bounded in a neighborhood of $a$. Can we then say that $f$ is analytic on $G$? I think what I need to do ...
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1answer
53 views
Analytic maps from upper half plane to itself
I know that the bilinear map $f(z)=\frac{az+b}{cz+d}, a,b,c,d\in \mathbb{R}, ad-bc>0$, maps the upper half plane onto itself. But is every analytic function that does so is of this form?
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0answers
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Writing $f(x)=h(x)\cdot(x-\alpha)^m$ when $\alpha$ is a root of multiplicity $m$
Let $f$ be a real function, and $\alpha \in \Bbb R$ a root of $f$ of multiplicity $m$. Under what conditions can I write $f(x)=h(x)\cdot(x-\alpha)^m$, where $h(\alpha)\neq 0$ ?
I know it's true if $f$...
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votes
1answer
30 views
Analytical functions equal at a point implies all derivatives are also equal at that point?
There is this problem that goes as follows:
If I have two analytical functions $f,g:I→R$
(where $I$ is an open interval), and I know that there is a point $a∈I$
where $f(a)=g(a)$ and also $f^{(k)}(...
1
vote
1answer
23 views
Real analyticity of harmonic functions
Let $u(x,y)$ be harmonic on a domain $D \in \mathbb C$, Can we say that $f(x)=u(x,0)$ is real analytic on $D \cap \mathbb R$ if it is non-empty?
Clearly $f$ is infinitely differentiable on the domain,...
6
votes
1answer
62 views
$f,g$ analytic on $I\subset \mathbb{R}$. If exist $a\in I$ such that $f=g$ and $f^{(n)}=g^{(n)}$then we have $f(x)=g(x)$ for every $x \in I$.
Problem
Let $f,g$ analytic on an open interval $I\subset \mathbb{R}$. If exist some $a\in I$ such that $f(a)=g(a)$ and $f^{(n)}(a)=g^{(n)}(a)$ for all $n \in \mathbb{N}$ then we have $f(x)=g(x)$ for ...
0
votes
0answers
24 views
Removable singularities of analytic functions on $\mathbb{R}^{n}$ ($n>1$)
Let $\mathcal{H}$ be a Hilbert space and let $f:(-1,1)^{n}/\{0\}\subset\mathbb{R}
%EndExpansion
^{n}\rightarrow \mathcal{H}$ be an analytic function. If $f$ is bounded,
does the limit $\underset{x\...
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votes
0answers
37 views
A paradox on real vs complex analyticity
We know two facts:
-A complex function is complex analytic if and only if it is once complex- differentiable.
-A real function is not necessarilty real analytic if it is once real-differentiable, ...
1
vote
0answers
15 views
PDE Analytic System
Can someone give me some help with the following problem? I think it relates with Cauchy-Kovalevskaya Theorem, or maybe some technique from its proof.
Let $u=(u_1, ..., u_N)$ and consider the ...
2
votes
0answers
18 views
PDE System and Analytic Functions
Please, could someone give me some help with this problem? I've struggling for days on it.
Let $a, b_{i j}$ be analytic functions defined in a neighborhood of
$0$ in $\mathbb{R}$ and $\varphi ,\...
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0answers
93 views
Dimension of the solution space of a real analytic system of equations
I am dealing with the following situation:
I have a real analytic system of equations, $F(x,t)=0$, with $F: [0,1]^n \times [0, \infty) \rightarrow \mathbb{R}^n $.
I know that only a single solution, ...
