Questions tagged [analytic-functions]
For questions about analytic functions, which are real or complex functions locally given by a convergent power series.
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Growth of analytic function at infinity, provided a bound in all but one directions.
Let's consider a function of a complex variable $f(z)$, with the following properties:
$f(z)$ is bounded by a constant as $z \to \infty$ away from the positive real axis ($z \notin \mathbb{R}^+$)
$f(...
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Analytic function in an open disc
Can someone help me with this question.
Or just can give a hint which theorem from complex Analysis to use to solve this. Really stucked!
Q. Let f(z) = $\sum_{n\ge0} a_{n}z^{n}$ be an analytic ...
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A question related to a complex inequality
Let $\Delta$ be the unit disc centered at $0$ in the complex plane and consider an analytic function $g:\Delta \to \Delta $ such that $g(0)=0$. I have proved the following claim (using Schwarz Lemma).
...
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A question related to the existence of the limit of a complex function
Let $\Delta$ denote the unit disc centered at $0$ in the complex plane. I'm interested in proving the following claim.
Assume that $g$ is a function that is bounded and analytic on
$\mathbb{C}\...
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For continuous $f(x)$, if $\lim_{x\to 0} f(x)/p(x)=0$ for every nonzero polynomial $p(x)$, then is the same true for nonzero analytic $p(x)$?
Let $f(x)$ be a continuous real function on a neighborhood of 0 in $\mathbb{R}$.
Suppose $f(x)>0$ for $x\neq0$, and suppose $\lim_{x\to0} f(x)/p(x)$ for every nonzero polynomial $p(x)$.
My question ...
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Hardy class of bounded analytic functions is Banach space
I need help with the following:
Let $H^{\infty}(\mathbb{D})$ denote Hardy class of bounded analytic functions on unit disc $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$. Prove that $$||f|| = \...
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A $\mathbb{C}$-valued function $f$ is real-analytic
Let $f:\mathbb{C}^n \longrightarrow \mathbb{C}$ be a $\mathbb{C}$-valued function.
When can it be said that the function $f$ is "real-analytic"?
Thanks in Advance.
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Conditions for compositions involving one analytic function to be analytic
Suppose that I have two functions, $f,g: \mathbb{C}\to\mathbb{C}$, of which $f$ is analytic. What can be proven of $g$ if $f \circ g$ is analytic? If $g \circ f$ is analytic?
Basically, what are the ...
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A question related to a complex function satisfying certain properties
I'm interested in finding all entire functions $g:\mathbb{C}\to\mathbb{C}$ such that $g(z+1)=g(z)$ and $g(z+i)=e^{2\pi}g(z)$, for all $z\in\mathbb{C}$.
My initial plan was to make use of Liouville's ...
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A question related to the existence of an analytic function [closed]
Is it possible to construct a function $g$ which is analytic on an open set containing the closed unit disc such that $g(z)=(\Im (z))^{2023}$, for all $z$ in the boundary of the unit disc?
I'm pretty ...
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Can Schwartz class functions be nowhere analytic?
By "Schwartz class functions" I will be referring to the functions of the Schwartz space on $\mathbb{R}$, that is, smooth ($\mathcal{C}^\infty$) functions $f : \mathbb{R} \to \mathbb{R}$ ...
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Singularities at Infinity implies constant or polynomial function - Proof?
I have been doing some leisurely reading on complex analysis, and have came across several texts that mention the following:
If an entire function has a removable singularity at infinity, then it ...
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Assume f(z) is analytic in a domain R and never vanishes, but for some $z_{o}$ in R: $|f(z_{o})|=min(|f(z)|)$ in R. How to prove that f is constant?
I tried solving this question by taking the derivative of f(z) = u(x,y) +iv(x,y). Then setting the derivative to zero at $z_o$. But doing this did not give any useful insight. I am generally confused ...
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Extensions of holomorphic functions
Firstly, denote by $P$ as the set of holomorphic functions that extend $f$, where $f: D \rightarrow \mathbb{C}$ is holomorphic, and $D$ is a domain in $\mathbb{C}$.
We have to show that $P$ has a ...
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Prove that $f$ is a polynomial function of $z$ [duplicate]
Given that $f$ is a holomorphic function on a domain $D \subset \mathbb{C}$, and $ \forall a \in D$ we have a $n_a \in \mathbb{N}$ such that $f^{n_a}(a) = 0$. We have to use this to show that $f$ is a ...
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Is there everywhere large real-part analytic function on the upper half-plane?
Let $\mathbb{H}$ be the upper half-plane of the complex plane. Does there exist an analytic function $A$ on $\mathbb{H}$ satisfying the inequality $\text{Re}(A(z))\ge \text{Im} (z)^{\alpha}$ for some $...
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Show two analytic functions are related by a constant
The problem I'm working on says this: Suppose $f, g$ are two analytic functions on an open set containing $\overline{D}$, where $D = \{ z \in \mathbb{C} : |z| < 1 \}$ is the open disc in $\mathbb{C}...
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two-times analytic continuation of the domain
Lets imagine that D - given domain (strip) on upper half plane H, and G - its image (connected) under mapping f with required properties. The image has curved side (not stright line, like, polygons!). ...
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Understanding a given proof for the identity theorem of analytic functions: Why do $f_1$ and $f_2$ agree on a neighbourhood of an accumulation point
My question is about a part of a given proof for the identity theorem of analytic functions. This part shows that two analytic functions are identical in a neighbourhood of an accumulation point of ...
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If a function $f$ is analytic and $f(x,y)=f(x,-y)=f(x+y,y)$ then the function depends on $y$ alone
Let $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a function such that $f(x,y)=f(x,-y)=f(x+y,y)$.
I need to show that if $f$ is an analytic function, then $f$ depends only on $y$
What I did was this:
Given ...
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Can the complex function $f(z) = \sqrt[n]{r^n - z^n}$ be analytically continued to fill the complex plane?
I have the complex function $f(z) = \sqrt[n]{r^n - z^n}$, where $z$ is a complex variable, $r$ is a positive real number, and $n$ is any non-zero real number. For any value of $n$ besides 1, this ...
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Is $\text{PSL}_2(\mathbb{R})$ the automorphism group of the upper half plane?
Let $f:\mathbb{H}\rightarrow\mathbb{H}$ be any analytic automorphism of the upper half plane $\mathbb{H}=\{z\in \mathbb{C}:\text{Im}z>0\}$. Then can it be written by the form
$$f(z)=\frac{az+b}{cz+...
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Are Characteristic Functions Analytic?
Let $X$ be a real random variable defined on a probability space $(\Omega, F, P)$. Define its characteristic function $\phi: \mathbb{R} \to \mathbb{C}$ by $\phi(t) = \mathbb{E}[e^{itX}]$ for every ...
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Is there any simple set of properties that uniquely characterizes differentiation in the space of complex functions?
The transformation of differentiation is a linear operator over the vector space of entire functions (call this space $\mathbb{C}^E.$) Is there any simple set of properties that uniquely determines ...
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What can one tell about the derivatives of a function which intersects a polynomial of degree $n$ at no more than $n+2$ points?
Consider an infinitely differentiable function $f:\Bbb R \to \Bbb R$ and define $\phi(n), n\in \Bbb N_0$ the maximum number of roots that the function $f(x)-p(x)$ can have for any polynomial of degree ...
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If any $n$-degree polynomial intersects $f$ at no more than $n+1$ then $f$ has all its derivatives positive.
Conjecture. An infinitely differentiable function $f:\Bbb R \to \Bbb R$ at some $x\in \Bbb R$ has all its derivatives nonzero with identical or alternative sign if and only for any polynomial $p$ the ...
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If $g(z)$ is analytic near $0$, and $g(z) = \exp(z\,g(z))$, then $g(z)=\sum_{n \geq 0}\frac{(n+1)^{n-1}}{n!} z^n$ near $0$
This is a question from the article "An Occupancy Discipline and Applications".
$g(z)$ is an analytic function near $0$. Suppose
$$g(z) = \exp(z\,g(z))$$
Then prove that, near $z=0$,
$$g(z) ...
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Proving that a complex function blows up near a pole [duplicate]
I am struggling to prove that if $z_0$ is a pole of order $k$ of $f(z)$, then $\lim_{z \to z_0} |f(z)| = \infty$, could anyone help point me in the correct direction?
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Can we find an analytic function $f(x)$ whose zeros are the product from the zeros of two given analytic functions?
Can we find an analytic function $f(x)$ whose zeros are the product from the zeros of two given analytic functions $g(x)$ and $h(x)$?
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Deducing that an analytic function $f$ in a simply connected domain $\Omega$ is the derivative of an analytic function $F:\Omega\to\mathbb{C}$
Let $\Omega$ be a simply connected domain, i.e. open and connected subset of $\mathbb{C}$ and $f$ be analytic in $\Omega$. I am trying to understand how you can deduce that $f(z) = \frac{d}{dz}F(z)$ ...
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Find all analytic functions $f(z) = u(x,y)+iv(x,y), z \in \mathbb{C}$ such that $\frac{\Re(f(z))}{\Im(f(z))} = \frac{u(x, y)}{v(x, y)} = g(x).$
Find all analytic functions $f(z) = u(x,y)+iv(x,y), z \in \mathbb{C}$
such that $$\frac{\Re(f(z))}{\Im(f(z))} = \frac{u(x, y)}{v(x, y)} =
g(x).$$ $g(x)$ is a function only of $x$, and doesn't depend ...
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The relationship between interior and exterior Riemann maps
Let $U\subset\widehat{\mathbb{C}}$ be a compact Jordan domain with $\partial U$ connected. Is there a relationship between the exterior Riemann map associated to $U$,
$$\varphi:\mathring{(\mathbb{D}^c)...
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Is the real part of a complex analytic function, real analytic?
Recently in Calculus 1 we were introduced to the concept of analytic functions (to be more exact, real analytic functions). At the same time I was familiar with the concept of complex analytic ...
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Modulus squared of derivative of analytic function [closed]
In my lecturers notes it is mentioned:
$(u_{x})^2 + (v_{x})^2 = |\frac{dw}{dz}|^2,$ where $w = u(x, y) + iv(x, y), \quad z = x + iy.$
But this isn't immediately obvious to me, if anyone could offer ...
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For which sequence do all associated power series converge? (describe the subset of $\Bbb R^\Bbb N$ "homeomorphic" to the analytic functions)
I am investigating a certain property of real smooth functions (though it is easily extended to complex analytic functions) which requires me to define a topology on $C^\infty(\Bbb R)$ that "...
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Sum of an infinite series [duplicate]
I want to calculate
$$ \sum_{k=1}^{\infty} \frac{k^2}{2^k} $$
Usually (assuming it was $k$ instead of $k^2$) I would consider this as a special case of the series $ f(x) =\sum_{k=1}^{\infty} k \cdot z^...
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Holomorphic Functions Accumulating Zeros
I want to prove that there is no holomorphic function $f(z)$ on the open unit disk that satisfies:
$$
f(\frac{1}{n}) = 2^{-n} \quad \quad \forall n\in\{2,3,4,\cdots\}
$$
I recall that if a holomorphic ...
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Left and right derivative of Fourier transform of a function supported in a half-line
Consider a function $f\in L^1(\mathbb{R})$, $f\geq0$, and let $\hat{f}$ be its Fourier transform,
\begin{equation}
\hat{f}(\omega)=\int_{\mathbb{R}}f(x)\,e^{-i\omega x}\,\mathrm{d}x.
\end{equation}
We ...
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Existence of an analytic function by estimate
Does there exist an analytic function on the unit disc s.t. $|f\left(\frac1n\right)−\frac{(−1)^n}{n^2}|<\frac{1}{n^3}$ for all $n\geq2$?
See my idea but I don’t know how to complete the proof. I ...
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Is it possible for a function to equal its Taylor series on a half interval
$\def\R{\mathbf{R}}$
Let $f:(-1,1)\to\R$ be infinitely differentiable.
Then it has a taylor series expansion $T(x)=\sum_{n=0}^{\infty}{a_nx^n}$ where $a_n=f^{(n)}(0)/n!$.
Let's say we're given that $T$...
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On the product of analytic functions
Let $f,g:(0,1) \to \mathbb{R}$ be analytic. It is well known that the product $h:=fg$ is then also analytic.
I would like to prove it by a using the following characterization of analyticity:
$h$ is ...
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Is $f(x,y) = \frac{x^2y^2}{x^2+y^2}$ analytic at the origin?
Is the function $f(x,y) = \frac{x^2y^2}{x^2+y^2}$ analytic at the origin?
Changing into polar coordinates $x= r\cos\theta, y=r\sin\theta$ gives $f = r^2 \sin^2\theta \cos^2\theta$. Hence, as a ...
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Decomposition of a given function: is the function necessarily be analytic?
I was reading the supplementary material of this paper and wonder if the decomposition of a given function $f(x, w) \approx \phi(x) a(w)$ is valid only when $f$ is analytic.
Can the condition of ...
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Is there a function analytic in $D(0,r)$ such that $(f(z))^2 = e^z + z$ for all $z\in D(0,r)$?
I'm stuck on the following problem:
Let $D = \{z\in\mathbb{C}:|z|<r\}$, $r>0$. Is there a function analytic in $D$ such that $(f(z))^2 = e^z + z$ for all $z\in D$? Look at the two cases:
$r = 1$...
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Problem in understanding the step in the proof of complex analysis
Let $K$ be a compact subset of the region $G$;then there are straight line segments $\gamma_1,\gamma_2,...,\gamma_n$ in $G-K$ such that for every function $f$ in $H(G),$
$$f(z)=\sum_{k=1}^{n}\frac{1}{...
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Inverse of a analytic function
Let $f$ be a map on the closed unit disc $\bar{\mathbb{D}}$ in $\mathbb{C}$ such that $f$ is analytic on $\mathbb{D}$ and continuous on $ \bar{\mathbb{D}}$. Can you tell under what condition will $f$ ...
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Question on Taylor's theorem with remainder
I am reading Tu's book "An Introduction to Manifolds" and I have a question related to Taylor's theorem with remainder (which is not related to this post: Taylor's theorem with remainder)...
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Branch cut of the function log(z + i).
I have a short question. I have the following function $f\left(z\right) = \log\left(z + i\right)$.
The question is, construct a branch cut such that $f\left(z\right)$ is analytic at $f\left(0\right) = ...
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What is the qualitative relationship between an analytic function and its analytic continuation on the complement of the originally defined domain?
Definition of analytic continuation which I am familiar with is that if $f_1$ is an analytic function in a connected open set $\Gamma_1$ then $(f_2, \Gamma_2)$ is an analytic continuation of $(f_1, \...
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Real Analytic function .
Let $f$ a real analytic function at $x=a$ i.e. having power series expansion about the point $x=a$. Can we say about radius of convergent of power series ? Like its equal to the distance form $a$ to ...
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