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 are quite different from the properties of functions over the complex numbers.
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Convolution of $\mathcal{C}^\infty$ is analytic
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$. Is the convolution (assumed to be well defined) defined as:
$$(f*g)(x) = \int_\mathbb{...
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Maximizing the radius of convergence around a point for an analytic function
Let $f:\Omega\longrightarrow \mathbb{C}$ be analytic, and let $z_0\in\Omega$ s.t.
$$f (z)=\sum_{n\geq 0} a_n(z-z_0)^n,\quad\forall \left|z-z_0\right|<r,$$
for some $r>0$, and some complex-valued ...
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restricting a function changes its singular points and analyticity?
Let define $f(z) = \frac{1}{z-2}$ for $z\in\mathbb{C}\setminus\{2\}$. Then it is clear that, $f(z)$ has singular point at $z=2$ (Namely pole of order 1 at $z=2$).
However, if I update the definition ...
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In complex number system, sin z and cos z are unbounded and periodic. But they are continuous also. How can that be possible?
I know that a continuous periodic function must be bounded because if a function is continuous and periodic, its graph will have to turn at certain points to reattain the values and hence, it cannot ...
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Regarding a Coin Toss Experiment by Neil DeGrasse Tyson, and its validity
In one of his interviews, Clip Link, Neil DeGrasse Tyson discusses a coin toss experiment. It goes something like this:
Line up 1000 people, each given a coin, to be flipped simultaneously
Ask each ...
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Analyticity of an "Integral" type function.
I found the following exercise in Conway's complex analysis.
Determine the region in which
$
f(z) = \int_{0}^{1} \frac{1}{{t - z}} \, dt
$ is analytic.
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Pointwise limit of sequence of holomorphic functions given constraint on their derivatives at the origin
Consider a sequence of functions $g_n(z)$. $n$ takes values on the natural numbers, and $z$ is a complex variable. For all $n$, $g_n(z)$ is guaranteed to be an analytic function of $z$ within a disk ...
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Analytic Solution to: $ n_t = C_1(n_x^2 + nn_{xx}) \text{ where }n(-L,t)=n(L,t)=0 \text{ and }n(x,0)=e^{-x^2} $
I am working with a nonlinear PDE and am looking for an analytical solution. I'm unsure how to figure out if a PDE has a known solution, so I figured that someone here may know.
This is the PDE:
$$ \...
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Why is a polynomial in $z,\bar z$ analytic iff it does not involve monomials $z^i\bar z^j$ with $j\ge1$?
I have encountered some note on complex analysis:
In particular, given a polynomial in the real variables x and y , with
complex coefficients, these properties tell us that the polynomial
is analytic ...
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Spatial analyticity for solutions of linear parabolic PDEs
I believe that the following result is true, but am having a hard time finding a suitable reference to convince myself:
Suppose $X:\mathbb{R}^n\to \mathbb{R}^n$ is analytic. If the smooth function $u:(...
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Smooth Riemannian metric is locally real analytic?
Let $U$ be an open subset of $\mathbb{R}^n$ and $g$ be a $C^\infty$ Riemannian metric on $U$. Given a point $x_0\in U$, does there exist a local neighborhood $x_0\in V\subset U$ and new coordinates ...
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Analytically continuing a function of two complex variables.
I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex ...
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Why does Jost function have analytic continuation on the $p$-plane?
I have question in the analyticity of Jost function on the $p$-plane. The chapter 12 of the book Scattering Theory by John R. Taylor states (p.218): The Jost function is defined as
$$
f_{l}(p)=1+\frac{...
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Computing the domain of analyticity of $f(z)=\sqrt{z^2-1}$
In this question, it is said that the domain of analyticity of the function $f(z)=\sqrt{z^2-1}$ over the branch $(0,2\pi)$ is $\mathbb{C} \setminus ((-\infty,-1) \cup (1,\infty))$.
My question: I ...
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Find the analyticity of |z|
I have been struggling finding the domain of analyticity in complex analysis, my teacher finds them with a drawing like the one I've attatched (in the pic they were asking for $\sqrt{z-1}=e^{1/2 \log(...
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