Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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10 views

determine type of singularities and compute residue of a function

Determine the type of singularities and residue of $$\frac{1}{\sin^2(z)}$$ For this problem, this is the way I approach this: we have : $$\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots$$ ...
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4answers
31 views

sum of the residues of all the isolated simgualrities

Prove that, for $n \geq 3$, the sum of the residues of all the isolated singularities of $$\frac{z^n}{1+z+z^2+\cdots+z^{n-1}}$$ is 0 Can someone show me how to do this problem. Thank you.
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1answer
18 views

Resources on Variants of the Clausen Functions

I am interested in locating more information about the Clausen functions. Specifically I am looking for the closed forms of the Gl-type (or Sl-type as they are sometimes called) and the alternating ...
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1answer
33 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the ...
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1answer
16 views

Reiterating the piecewise-and-uniform-limit operation

Probably a hopeless question, but: Let $C$ be the class of constant functions $f$: $[a, b]\longrightarrow\mathbb{R}$. Let $\mathcal{U}(\mathcal{P}(C))$ denote the class of uniform limits of piecewise ...
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0answers
15 views

Question about a proof in Clapp's paper, “On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem”

I have this: And I want to understand this proof: I don't understand the choice of $f(p)$ why it is sufficient to prove that $\lim_{p\rightarrow2^*} f(p)=f(2^*)$ since we see that ...
2
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1answer
19 views

Let $f$ be differentiable at every point of some open ball $B(a)$ in $\mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then prove $D_k f(a)=0$.

If $f:\mathbb R^n \to \mathbb R$ is a function differentiable at every point of some open ball $B(a)$ with center $a\in \mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then how to show that all ...
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2answers
14 views

$f'(x;y)=0$ for every $x$ in an open convex set and for every vector $y$ ; then to show $f$ is constant on $S$

Let $f:\mathbb R^n \to \mathbb R$ be a map , $S$ be an open convex set in $\mathbb R^n$ such that for every $x \in S$ and $y \in \mathbb R^n$ , $f'(x;y)$ exists and equals $0$ ; then how to show that ...
0
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0answers
20 views

The boundary integral equation

in which case we use the single layer potential and the double layer potential for the Laplace equation ? \begin{eqnarray}\tag{1} \Delta u = 0 \; \mathbb{R}^2\backslash\omega\\ u \to 0 \; at \; ...
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1answer
17 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
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0answers
22 views

$e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$

I have to show that $e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$ holds for all positive integers k and m whenever $m>4\cdot k\cdot log(k)$. I replaced $(1-\frac{1}{k})^m$ with ...
5
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1answer
24 views

if $\operatorname{Res}_{z_0}f = 0$, then $f$ has a primitive in some deleted neighborhood of $z_0$

Let $z_0$ be an isolated singularity of $f$. Prove that if $\operatorname{Res}_{z_0}f = 0$, then $f$ has a primitive in some deleted neighborhood of $z_0$ I know that if we assume $f$ has a ...
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0answers
41 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
3
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1answer
60 views

Is it possible to develop Analysis solely from Peano's axioms

...and a few definitions on the way? When I studied Calculus using Spivak's book It was clearly shown that, in order to prove some fundamental theorems (intermediate value theorem being one of them), ...
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3answers
20 views

Fixpoints and continuity

I don't understand why this is true: If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$ I don't understand why such a point exist. Why is there not ...
5
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2answers
41 views

On any continuous map $f:S^1 \to \mathbb R$

Let $f:S^1 \to \mathbb R$ be any continuous map , where $S^1$ is the unit circle in the plane . Let $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ ; then how to prove $A$ is uncountable , or ...
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1answer
32 views

How to know which notion of convergence to use when proving density of a subspace

My question might be a little vague, but is there a way to know which type of convergence (i.e pointwise, uniform) to use when proving that a subspace is dense in a certain space. For example if we ...
1
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1answer
13 views

Formula to calculate adjusted score within range

I'm trying to come up with a formula to combine two different score types, where the second score will weight the first score without exceeding the upper or lower ranges of that score. In the interest ...
3
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1answer
68 views

$|f(x)-f(y)| \geq \frac{|x-y|}{2}$

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $|f(x)-f(y)| \geq \frac{|x-y|}{2}$ then prove $f$ is onto. I can prove it just using IVT, but looking for some short solution which ...
1
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0answers
46 views

Under what conditions is this true: $\lim_{r \to 0} \frac{1}{r} \int_{0}^{2\pi} f(r,x) dx = 2\pi f(0,0)$

I will like to know under what hypothesis the following is true, and maybe a sketch of the proof. I saw it in a solution of an exercise. In this exercise, $f$ was harmonic, but I don't know if that is ...
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1answer
29 views

Another question on finiding special kind of power series [on hold]

Let $\sum a_nx^n$ be a real power series with finite positive radius of convergence $R$ ; then does there exist a non-constant real sequence $\{b_n\}$ such that $\sum b_nx^n$ is convergent for at ...
4
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1answer
32 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
2
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0answers
24 views

Need help with this calculus inequality

I'm currently studyding Classical and Multilinear Harmonic Analysis. Vol. 1 by Camil Muscalu, Wilhelm Schlag. I need to verify following calculus inequality(Eq. 9.27, at page 255) ...
1
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1answer
17 views

Continuity of a parametrized surface integral of a sobolev function

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain and let $v\in H^1(\Omega)$. Furthermore, let $S=(0,T)$ denote a time interval and let $s\in ...
7
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2answers
97 views

Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?

I checked these What is the difference between square of sum and sum of square? Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. It is easy to see $p$-th power ...
1
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2answers
31 views

Find supremum and infimumm of a set with two variables

$$A= \left\{\frac{m}{n}+\frac{4n}{m}:m,n\in\mathbb{N}\right\}$$ Since $m,n\in \mathbb{N}$, infimum is zero because $m,n$ both are increasing to infinity. Then the supremum is $5$ when $m,n$ are ...
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0answers
14 views

PDE reduced to ODE Uniqueness??

Could you please help me with the following problem. As a first help, I know the solution of the following ODE: \begin{align} j_1(t)[r \log(j_1(t)) + \beta] &= j_1'(t) \\ \nonumber j_1(T) ...
0
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1answer
21 views

Prove that regular curves are locally invertible

Consider the function $F = (F_1, F_2)$ from $I = (a, b) \subset \mathbb{R}$ to $\mathbb{R}^n$ (without loss of generality, assume $n = 2$). Suppose $F$ is differentiable (i.e $F_1' = f_1$ and $F_2' = ...
1
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0answers
19 views

Existence and uniqueness solution of a differential equation

If I have the following equation: $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'e^{\int_0^t\int_0^1G(|r-r'|)y(s,r')dr'ds}-y(t,r)$ $ y(0,r)=a(r)$ where $G:\mathbb{R}^+\to\mathbb{R}$ is ...
1
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0answers
31 views

Let S be a set. Let X be the set of bounded functions S $\times$ S to R with the supremum metric. Is the subset T of bounded metrics closed in X?

Let S be a set. Let us say that a metric d on S is bounded if there exists a real number R such that d(x, y) ≤ R for all x, y ∈ S. Let X be the set of all bounded functions S × S → R regarded as a ...
4
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0answers
76 views

Integral $\int z^2\Re(J_1(z))dz$=$\int y^{3/2} \Re \left[\frac{1}{\sqrt y} (1-e^{-y})\right]dy$

Hi I am trying to simplify and calculate the integral below. $$ I=\int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} ...
0
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1answer
14 views

Inverse function for a sort of negative binomial distribution

I am trying to find the inverse function of $f(p) = \sum_{k=0}^{6}{\binom{6-H+k}{k} p^{7-H} (1-p)^k}$, where $0 \leq H \leq 6$ is a constant integer. Any ideas on how to do this? Or perhaps equally ...
2
votes
2answers
79 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
1
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1answer
47 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
1
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3answers
76 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
1
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1answer
21 views

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
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0answers
35 views

Prove that a bijective entire function is uniformly continuous

Let $f$ be a bijective entire function. Prove that $f$ is uniformly continuous. I want a direct proof of this without using the fact that $Aut(\Bbb C)$ is the collection of linear polynomials ...
1
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1answer
38 views

Tough problem on sum of infinite series [on hold]

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
0
votes
1answer
31 views

X and Y are compact metric spaces. Show that X $\times$ Y is compact

$X$ and $Y$ are compact metric spaces with metrics $d_X$ and $d_Y$. $X \times Y$ is a metric space with the metric $d((x, y),(x , y' )) := \max\{d_x(x, x' ), d_y (y, y' )\}$. I want to show that ...
0
votes
0answers
16 views

Is the closure of every bounded convex set in $\mathbb R^n (n>1)$ homeomorphic to a closed ball ? [on hold]

Is the closure of every bounded convex set in $\mathbb R^n (n>1)$ homeomorphic to a closed ball ?
2
votes
1answer
58 views

Is a bijective entire function uniformly continuous?

Let $f$ be an entire function such that $f$ is bijective. Is then $f$ uniformly continuous? I am thinking on this when trying to compute the analytic automorphisms $Aut(\Bbb C)$. I know that ...
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0answers
31 views

Prove that $F_1$ and $F_2$ are continuous and that $\int_{\gamma_1}F_1(z) dz = \int_{\gamma_2}F_2(z) dw$

Let $\Omega_1, \Omega_2 \subseteq \mathbb{C}$ and let $\gamma_1: [a,b] \to \Omega_1$, $\gamma_2: [c,d] \to \Omega_2$ be paths. Let $f$ be a continuous function defined on $\gamma_1 \times \gamma_2$ ...
7
votes
0answers
122 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
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0answers
13 views

Basic examples of functions in Hörmander class

The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with $$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq ...
7
votes
1answer
127 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
5
votes
1answer
180 views

Proving existence of at least one root

The function $f:\mathbb{R}\to\mathbb{R}$, is continuous and $a>0$. How can I prove that there is at least one root of this equation: $f(x)=f(\sqrt{|x^2-a|})$
1
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2answers
48 views

Finding all complex roots of this equation

So i have this equation: $z^5-4z^4+11z^3+12z^2-42z+52=0 \text{ for }z\in\Bbb{C}$ One root is: $z=1+i$ That gives us also the 2nd root. $z=1-i$ But i am stuck with how to get other 3. I thought i ...
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votes
1answer
34 views

prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$

$\displaystyle\lim_{x \to a}f(x) = \lim_{x \to a}g(x)$ exist and $\displaystyle\lim_{x \to a}(h_1(x)g(x)+h_2(x)f(x))$ exist prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$ exist I would ...
3
votes
1answer
25 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
1
vote
1answer
79 views

How to rigorously establish this limit of sums

Assuming that $$\lim_{n}\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)=\int_{\mathbb{R}} f(u)g(u)\mathsf du,$$ (where $f$ is $C^2$ and $g$ and $g_n$ are probability distribution functions) I ...