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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20 views

Upper bound for the ratio of Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex and $z$ is a positive real number. Do you know any results about it? Thank ...
2
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0answers
46 views

Prove there exists a infinitely differentiable function whose value of partial derivatives of all orders at $0$ is a given function

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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0answers
38 views

Proof that $C^{\infty}_0$ is dense in $W^{1,p}(\mathbb{R}^n)$

I have taken $u \in W^{1,p}(\mathbb{R}^n)$ and a countable cover of $\mathbb{R}^n$ by open balls of increasing radius. I was hoping to mollify and use a partition of unity to deduce that ...
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6answers
63 views

check if the series $\sum_{k=1}^{\infty}\left(\frac{3k+1}{\sqrt{2k-1}}\right)^k$ converges

Check if the following series is convergent or divergent: $$\sum_{k=1}^{\infty}\left(\frac{3k+1}{\sqrt{2k-1}}\right)^k$$ This series should be divergent too, one way to see this is the comparison ...
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2answers
89 views

How would one solver this integral? [closed]

$$\int {dx \over (1-x^4)^2}$$ Tried partially , didnt work for me, and couldnt find a suitible substituition.. Any ideas?
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3answers
39 views

$W^{1,\infty}$ function that cannot be approximated by Smooth Functions

I am trying to find a function $u \in W^{1,\infty}$ that cannot be approximated by smooth functions. It seems like it should be an easy construction but I am blanking. Thanks.
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1answer
35 views

How to prove a space is a dual space?

How does one go about proving that a space is a dual space? The only thing I can think of is to prove that the space is isomorphic to a dual space. Is there a better way to do this? Thank you
2
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2answers
37 views

Show that $\widehat{f}(n)$ is zero for odd $n$

The following problem is from Stein´s Introduction to Fourier analysis: Suppose that $f(\theta + \pi)=f(\theta)$ for all $\theta \in \mathbb{R}$ Show that $\widehat{f}(n)$ is zero for odd $n$. My ...
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0answers
24 views

Derive the equation of first variation for a flow of a vector field.

This is a problem from Susan Colley's Vector Calculus. I have trouble understanding the solution to it. Problem: Derive the equation of first variation for a flow of a vector field. That is, if ...
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1answer
27 views

Writing a Fourier series of a $2\pi$-periodic function.

This problem was taken from Stein's Introduction to Fourier analysis, and it goes like this: Let $f$ be a $2\pi$-periodic Riemman integrable function defined on $\mathbb{R}$. Show that the Fourier ...
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3answers
39 views

Problem with finding some three points for non-monotonic function

Let $f:(a,b|) \rightarrow \mathbb R$ be a function which is not monotonic. I wish to prove that there are numbers $x<y<z$ from $(a,b)$ such that $ f(x)>f(y)<f(z)$ or ...
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0answers
22 views

How to evaluate the coefficient of power series?

For some reason(or trick), I need to calculate something like the coefficient of $X^6$ in $f(X)=\frac1{(1-X)(1-X^2)}\times\frac1{(1-X)(1-X^2)(1-X^3)(1-X^4)}$ evaluated as power series. How should I ...
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1answer
34 views

Fourier transform, circular symmetry

I need to compute the two-dimensional Fourier transform of a function with circular symmetry: $$ \int dx dy\, \frac{e^{i (k_x x+k_y y)}}{((z'-z)^2-t^2+x^2+y^2)((z'+z)^2-t^2+x^2+y^2)} $$ For ...
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1answer
50 views

Is the function $f: (1, \infty) \to \mathbb R$ defined as $f(x)=\sum_{n=1}^\infty n^{-x}$ continuous ?

Is the function $f: (1, \infty) \to \mathbb R$ defined as $f(x)=\sum_{n=1}^\infty n^{-x}$ continuous ? I know that for each $n \ge 1$ , the function $g:(1,\infty) \to \mathbb R ; g(x)=n^{-x}$ is ...
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votes
1answer
23 views

Distance between a density operator and a pure quantum state.

Given density operators $\rho_1$ and $\rho_2$ and a pure quantum state $|\psi>$. It is promised that $|\psi>$ is in only one of the given density operators. How to find which density operator ...
5
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3answers
166 views

Concerning series of positive real numbers whose terms are decreasing and tending to $0$

Let $\{a_n\}$ be a decreasing sequence of positive real numbers such that $\lim_{n \to \infty} a_n=0$ and $\sum_{n=1}^{\infty}a_n= \infty$ ( for eaxmple , like $a_n:=\dfrac 1n$ ), then is it true ...
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4answers
2k views

Is it necessary for one to understand analysis?

Is it necessary for one to understand analysis in order to pursue a career in mathematics? Basically, I am very weak at analysis. But the problem is that most of the topics listed in the syllabus ...
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2answers
40 views

If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$

Just wanted to confirm that this is a correct solution: Proof: Suppose $f(x_0) > 0$ for some $x_0 \in [a,b]$. Then, by continuity of $f$, for $\epsilon < f(x_0)$, there exists $\delta > 0$ ...
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0answers
11 views

Show that $T(\partial \mathbb{H^n} \cap U)=\partial \mathbb{H^n} \cap V$ where T is a $C^1$ diffeomorphism

Let $U$ and $V$ be open subsets of $\mathbb{H^n}$ and $T: U \to V$ be a $C^1$ Diffeomorphism. Show that $T(\partial \mathbb{H^n} \cap U)=\partial \mathbb{H^n} \cap V$. where $\mathbb{H^n}=\{x \in ...
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0answers
37 views

Riemann sum limit of function of two variables

I was wondering which of the two following statements was correct and why: $\lim_{N\to\infty} \frac{1}{N}\sum_{i=1}^N f(\frac{i}{N},g(\frac{i}{N}))=\int_0^1 f(u,g(u))du$ $\lim_{N\to\infty} ...
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1answer
28 views

Convergence test via integral

I've got to the problem of testing convergence using the integrals on $$ \sum _{n=1} ^{\infty} \arcsin \left( \frac{1}{\sqrt{x}} \right) $$ Our theory says: Consider an integer $N$ and a ...
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0answers
15 views

Lenght of graph of a function

I have found a simple problem. I want to know, how long the curve of function $e^x$ is going to be. For that, I found out, that I can use definite integrals, so for lenght from $a$ to $b$ it is $$ l ...
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2answers
33 views

a question about functional analysis conclusion,and I am not sure whether it is true or not?

we have $R^n$,$R^m$ spaces, suppose open set $O_{1}\subset R^n $ and $O_{2}\subset R^m$, $f:O_{1}->O_{2} $ is k-times differentiable$(1<=k<=\infty)$,then at $x_{0}\in O_{1}$,$rank(f)(x_{0})$ ...
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1answer
26 views

Closed Covering of a Compact Space

I was having a little question on my mind for which I found no solution (by myself) even though it seems like something that I should be able to know. What I was wondering is this: For any compact ...
2
votes
1answer
36 views

Definite integral with trigonometric functions [duplicate]

I have problem finding, how to solve this integral $$ \int _0 ^{\frac{\pi}{4}} \frac{3 \sin x + 2 \cos x}{2 \sin x + 3 \cos x}dx $$ This I can rewrite as $$ \int _0 ^{\frac{\pi}{4}} \frac{12 \sin ^2 ...
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1answer
18 views

Why are they homeomorph?

Let us have S = {$x=(x_1,x_2,x_3) \in \mathbb{R^3}: x_1^2 + x_2^2 + x_3^2 = 1$} and $a = (0,0,1)$. Show, that S\{a} and $\mathbb{R^2}$ are homeomorph. I know I should use the definition of ...
3
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2answers
61 views

How to find the limit properly?

I would like to find or solve the limit of: $$\lim_{n \to \infty} \frac{80^{(n+1)/4}}{37^{(2n+3)/4}}$$ My idea was somehow non-intuitive: $$\lim_{n \to \infty} \frac{80^{(n+1)/4}}{37^{(2n+3)/4}} ...
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1answer
20 views

the position vector $x(t_0)$ is orthogonal to the velocity vector $x'(t_0)$ if $x(t_0)$ is the point on the image of $x$ closest to the origin .

Let $x(t)$ be a path of class $C^1$ that does not pass through the origin in $R^3$. If $x(t_0)$ is the point on the image of $x$ closest to the origin and $x'(t_0)\neq 0$, show that the position ...
11
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3answers
153 views

Stuck on crucial step while computing $\int_{- \infty}^{\infty} e^{-t^2}dt$

This is a not mandatory exercise I am struggling with from my Analysis II Class, at the very end of it I am supposed to compute $$\int_{-\infty}^\infty e^{-t^2}dt \tag{*}$$ The most famous way to do ...
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0answers
19 views

Show that an open subset $U \subset \mathbb{H^n}$ is open in $\mathbb{R^n}$ iff $U \cap \partial\mathbb{H^n}=\phi$

Consider the $n-$dimensional closed half-space $\mathbb{H^n}=\{x \in \mathbb{R^n}|x_1 \le 0 \}$ and let $\partial \mathbb{H^n}=\{x \in \mathbb{H}|x_1=0\}$ be its boundary. Show that an open subset $U ...
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0answers
22 views

Weird behavior of Non-Lebesgue measurable subset of the Smith–Volterra–Cantor set and pointwise convergence of a sequence of simple functions

I came across this (seemingly?) weird behavior of a sequence of simple functions: Let $E$ be the Smith–Volterra–Cantor set and $m: \operatorname{Leb}(\Bbb{R}) \to [0, \infty]$ the Lebesgue-measure. ...
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0answers
24 views

Can someone show me how to calculate this inequality?

$\sum_{2^j\le 1/2^k|y|}2^{j+k}|y|+\sum_{1/2^j|y|\le 2^j\le 2R/|y|}1+\sum_{2^j\ge 2R/|y|}(2^{j+k}|y|)^{-1}$ is less than equal to $|\log R|+|k|$. It's from Grafakos' Classical Fourier Analysis p.378. ...
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1answer
23 views

Change the order of integration:$\int_{0}^{1} dx \int_{0}^{1}dy \int_{0}^{x^2+y^2}f(x,y,z)dz$

$$\int_{0}^{1} dx \int_{0}^{1}dy \int_{0}^{x^2+y^2}f(x,y,z)dz$$ in the order : $$\int dz \int dx \int f(x,y,z)dy$$ I don't think it needs to be divided into two seperate integrals, but the professor ...
1
vote
1answer
30 views

Topology and Monotone Convergence Theorem

I'm looking on the Monotone Convergence Theorem and asking myself whether it is the property of ANY order topology induced by some total order that is dedekind complete ( basis are the open-intervals ...
1
vote
1answer
36 views

Prove the triangle inequality for d(x,y) = min(|x−y|,1−|x−y|)

Let X be the set [0,1). Define a non-standard metric on X as follows: For two numbers x,y ∈ X, take d(x,y) = min(|x−y|,1−|x−y|). Show that this is a metric. In order to show this is a metric, I need ...
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2answers
149 views

Covering the plane by squares!

$K_n$ is a sequence of squares of area $a_n$. Show that if $\sum_{n=1}^\infty a_n=\infty$ then we can arrange the squares $K_n$ to cover $\mathbb{R}^2$. Comments: -obviously we can suppose that ...
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3answers
33 views

If $f$ is a uniformly continuous function show that $g = f(x) - f(y)$ is uniformly continuous on all of $\mathbb{R}^2$

Problem statement: Let $f: \mathbb{R} \to \mathbb{R}$ be a uniformly continuous function on $\mathbb{R}$ and let $g: \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x) - f(y)$. Then $g$ is uniformly ...
2
votes
1answer
36 views

Product of two smooth functions that is identically zero

assume I am given with two smooth functions $f,g:[a,b]\to\mathbb{R}$ for which: $$ f(x)\cdot g(x)=0 $$ for every $x\in[a,b]$ . Does this imply $f\equiv 0$ or $g\equiv 0$ on some subinterval ...
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0answers
11 views

Proving that the gradient of $f$ at $x_{0}$ is equal to the vector where the columns are the partial derivative of the function at $x_{0}$

I proved the corollary written below, However I couldn't show that $\frac{1}{h}R(x_{0}+he_{i})=o\left(\left|x_{0}+he_{i}\right|\right)$, I wrote (Complete) to the place I should have proven it. Can ...
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0answers
72 views

Career Advice: I love Abstract Algebra and Analysis… What should I do?

I love discrete things as much as continous things... Does a PhD in Abstract Algebra necessarily mean abandoning continuous objects, integral etc.? Or Does a PhD in Analysis necessarily mean ...
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2answers
51 views

If an infinite series is divergent, after rearrangement can it be convergent?

I was studying regarding Riemann rearrangement theorem which was regarding conditionally convergent series. Now I am wondering, if an infinite series is divergent, does there always exist some ...
3
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1answer
97 views

If $a_i\geq 0,$ $\sum\limits_{n=1}^\infty a_n$converges, prove $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}$diverges.

If $a_i\geq0,a_n\neq0,$$\sum\limits_{n=1}^\infty a_n$converges, prove $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}$ diverges. I have known that if $\sum\limits_{n=1}^\infty a_n$converges, ...
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1answer
72 views

Calculating $f(x,y)=3xy +x^2$ on the unit circle

$f:\mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \mapsto 3xy +x^2$ $D:=\{(x,y)\in \mathbb{R^2}: x^2+y^2 \leq 1\}$ $\iint_Df(x,y)dxdy=\int_0^1 ...
0
votes
1answer
20 views

Limit of quotients implies limit of roots

If $a: \Bbb N\to (0, \infty)$ is a sequence with $\lim a_{n+1} / a_n = L < \infty$, then it has to be shown that $\lim a_n)^{1/n}$ = L. Has anybody a hint for me? I tried using the definition ...
5
votes
3answers
500 views

Improper integral of a rational function!

Find the value of the integral $$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx$$ I tried the substitution $x=t^5$ to obtain $$\int_0^\infty \frac{5t^6}{1+t^{10}}dt$$ Now we can factor the denominator to ...
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0answers
24 views

Packing Problem

Hey I'm trying to solve a problem of figuring out how many rectangles of a certain size would fit within a triangle and a trapezium, is there a formula that can be used for this? Say I had a ...
1
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0answers
32 views

Linear mapping $l$ vanishes at $x_0$ faster than $|x-x_0|$ if and on only if $l=0$

I need proof verification, any feedback will be appreciated, thank you. I have to prove that $l\in Hom(\mathbb{R}^{n},\mathbb{R}^{m})$ vanishes at $x_0$ faster than $|x-x_0|$ if and only if $l=0$. If ...
1
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1answer
23 views

Dominated convergence and fundamental lemma of the calculus of variation

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
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2answers
57 views

For two path-connected homeomorphic spaces, is there a homeomorphism with a prescribed value at a point? [duplicate]

Let $X$, $Y$ be homeomorphic path-connected topological spaces. Is is true that for any pair $x\in X$, $y\in Y$ a homeomorphism can be chosen such that $h(x)=y$?
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1answer
22 views

Prob. 3, Sec. 2.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by $$\Vert x ...