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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-1
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2answers
46 views

Find the value of the integral $\int_{-\infty}^{\infty} \frac{1}{1+(a-y)^2} \frac{1}{1+y^2} dy$

Can someone help me with the integral : $\int_{-\infty}^{\infty} \frac{1}{1+(a-y)^2} \frac{1}{1+y^2} dy$
0
votes
0answers
51 views

Integral in Gauß Bonnet theorem

I just read a text about the Gauß Bonnet theorem. If I have a function $f:\Omega \rightarrow M$ defining a two-dimensional manifold $M$ with a boundary that is parametrized by a curve $c: I ...
1
vote
0answers
28 views

Evans PDE derivation of solution for Poisson's equation - integration by parts

In Evans' PDE text, when proving the solution to Poisson's equation, when he integrates by parts he takes the inward pointing normal, whereas in the integration by parts formula in the appendix, it ...
1
vote
2answers
36 views

Implicit function theorem in comparative static problem

The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = ...
0
votes
0answers
9 views

Coverings vs. $\delta$-separation

Let $E$ be a compact subset of $\mathbb{R}^n$. Why is it that the smallest number of balls of radius $\delta$ required to cover $E$ is comparable to the largest number of points possible in a ...
3
votes
1answer
39 views

Identity involving “the distance to the nearest integer” function

It's a problem of an exercise list: I want to prove that, if $||x||$ = the distance to the nearest integer to $x$, then: $$\sum_{n=1}^{\infty}4^{-n}||2^nx||=||x||(1-2||x||)$$ is true for every $x ...
5
votes
0answers
50 views

Three-colored plane

Let $\Pi$ is cartesian plane with the usual topology. $A, B, C$ are pairwise disjoint subsets $\Pi$, $A\cup B\cup C = \Pi$. Each of these sets is dense in $\Pi$: $\overline{A}=\Pi$, and ...
1
vote
1answer
22 views

f a function from (a,b) to reals and is continuous at c in (a,b) where f(c) is not 0. Show there exists a d>0 st |f(x)|>|f(c)|/2 for all x st |x-c|<d.

$f$ a function from $(a,b)$ to reals and is continuous at $c$ in $(a,b)$ where $f(c)$ is not $0$. Show there exists a $d>0$ st $|f(x)|>|f(c)|/2$ for all $x$ st $|x-c|<d$. I've been playing ...
0
votes
0answers
31 views

Generalize an average to a sum

Any help would be appreciated! Thanks so much!
2
votes
0answers
62 views

homogeneity of non-algebraic function [closed]

Suppose one has a system $$ \dot x_1 = f(x_1) $$ where merely a few properties are known. Properties are $f$ is nonlinear, Lipschitz continous, bounded, $f(0) = 0$ and $f \geq 0$. Is it possible to ...
1
vote
2answers
45 views

show that a function is constant

Let $h,g\in \mathbb{R}^{R}$ s.t $\forall x,y\in\mathbb{R}$ we have $(h(x)-h(y))(g(x)-g(y))=0$ show that $h$=constant or $g$=constant we have to show that for all $x\in \mathbb{R}$ ...
7
votes
1answer
41 views

Find, $\sum_{n=-\infty}^\infty f(n)$

For any integer $n$ define $k(n)=\frac {n^7} 7+\frac{n^3}{3}+\frac {11n}{21}+1$ and $$f(n)=\begin{cases} 0 & \text{if $k(n)$ is an integer} \\ 1/n^2 & \text{if $k(n)$ is not an an ...
-1
votes
3answers
33 views

The curve $x^3-y^3=1$ is asymptotic t the line $x=y$. Find the point n the curve farthest from the line $x=y$. [duplicate]

The curve $x^3-y^3=1$ is asymptotic t the line $x=y$. Find the point in the curve farthest from the line $x=y$ This is just need of further details in this ...
0
votes
1answer
23 views

Write down a partial differential equation satisfied by the function $g$.

Let $f$ be a differentiable function of one variable and let $g$ be t he function of two variables given by $g(x,y)=f(ax+by)$,where $a,b$ are fixed non-zero numbers. Write down a partial differential ...
0
votes
0answers
28 views

Continous extension of the function $P$ - how to make it?

Set $D:=\left\{z\in\mathbb{C}: \lvert z\rvert <1\right\}$ and define $P\colon D\times D\to\mathbb{R}$ by $$ P(x,y):=\begin{cases}\frac{1-\lvert x\rvert^2}{\lvert x-y\rvert^2}, & ...
1
vote
1answer
43 views

Why is $\sum_{m=0}^k \frac{k^2+k}{2} = \frac{(k^2+k)(k+1)}{2}$?

I recieved an answer like this : $S= \displaystyle \sum_{m=0}^k \dfrac{k^2+k}{2} - k\displaystyle \sum_{m=0}^k m + \displaystyle \sum_{m=0}^k m^2=\dfrac{(k^2+k)(k+1)}{2}-\dfrac{k\cdot ...
0
votes
1answer
18 views

$h=0 \text{ over } [u,+\infty)$

let $h: \mathbb{R}^{*}_{+}\to \mathbb{R}$ decreasing such that $h(u)=0,\quad u> 0$ and $f(x)=x \ h(x)$ is increasing Show that: $$h=0 \text{ over } [u,+\infty)$$ let $x\in ...
0
votes
5answers
58 views

Why isn't $\sum_{m=0} ^{k}\frac{k^2 + k -2mk +2m^2} {2}=\frac{(-3k+1)(2k^2+1)}{12}?$

The right equation is $$\sum_{m=0} ^{k}\frac{k^2 + k -2mk +2m^2}{2}=\frac{k(k+1)(k+2)}{3}?$$ In my calculation $$\sum_{m=0} ^{k}\frac{k^2 + k -2mk +2m^2}{2}=\frac{(-3k+1)(2k^2+1)}{12}.$$ I don't ...
1
vote
1answer
112 views

Integration of $\int_{-\infty}^{\infty}\frac{\cos x}{a^2+x^2}dx$ [duplicate]

I'm trying to find the integral $$\int_{-\infty}^{\infty}\frac{\cos x}{a^2+x^2}dx$$ Wolfram alpha says this is $$\frac{\pi e^{-a}}{a}$$ But how do you get this result? I tried using partial ...
0
votes
4answers
56 views

Why is $ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)$? [closed]

The problem is: Why is $$ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)\;?$$
0
votes
2answers
35 views

Equivalent matrics

Let $ (X,d) $ be a metric space and let $f:[0,\infty)\to [0,\infty)$ be a continuous function with the following properties: (i) $ f(x)=0 $ iff $x=0$. (ii) $ f(x)\leq f(y) $ if $ 0\leq x\leq y $. ...
0
votes
1answer
12 views

The uniform convergence of the difference quotient of a smooth compactly supported function

This question is the same one as the question found here: Explanation on a proof of a property of mollifiers However, the answers given just confuse me more. The question asked in that thread is ...
0
votes
1answer
30 views

Assume $\arg(z_1)-\arg(z_2)=2n\pi$. Show that this implies $|z_1+z_2|=|z_1|+|z_2|$

I am really lost here. Can anyone please give me a hint or two?
1
vote
3answers
35 views

Assume $|z_1+z_2|=|z_1|+|z_2|$. Show that this implies $\arg(z_1)-\arg(z_2)=2n\pi$

The hint I am given is that the relationship of $|z1||z2|$ implies $\arg(z1)-\arg(z2)=2n\pi$ is to be used somewhere. I think the only way this can be done is to square it but after that I'm not ...
0
votes
1answer
26 views

Assume $\theta_1-\theta_2=2n\pi$. Prove that $\text{Re}(z_1 \bar z_2)=|z_1||z_2|$

I proved the other way around already but I can't seem to prove this way. Both $\theta_1$ and $\theta_2$ are the arguments of $z_1$ and $z_2$ respectively. Can anyone help me out here? Edit: The ...
-3
votes
1answer
44 views

question about the euclidean metric

Let $E = [0,1] \cup \{2\}$ and $d$ the euclidean metric. How to find all the sequence $(x_n) _{n\geq1}$ include in E that $x_n$ converge to 2?
2
votes
1answer
80 views

Proving that a function is Hölder-continuous

Let $\alpha\in(-1/2,0)$ and $x\in (0,1)$. Define the function $$f(x) = \int_x^1 z^{\alpha-\frac{1}{2}} (z-x)^{\alpha}dz.$$ I have the feeling that $|f(x)-f(y)| \leq C|x-y|^{1/2}$ or any other power ...
2
votes
3answers
106 views

Integrate rational function with multiple complex roots

I want to integrate $$ \int_{-1}^1 \frac{x^2}{(1+n^2x^2)^2} dx. $$ By WolframAlpha I know the solution is $$ \int_{-1}^1 \frac{x^2}{(1+n^2x^2)^2} dx = \frac{\arctan(n) - \frac{n}{n^2+1}}{n^3}. $$ Do ...
2
votes
1answer
54 views

$\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$

Determine if the following limits exist $$\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$$ note that $$\frac{1}{x}-1 <\lfloor \frac{1}{x}\rfloor \leq \frac{1}{x}$$ $$1-x ...
1
vote
2answers
40 views

Metric equivalence

I have that $E=[0,1]$ and $d'(x,y)=|\sqrt{x}-\sqrt{y}|$ and i want to prove that $d'$ and the usual metric $d(x,y)=|x-y|$ are not equivalent in the metric sense. I proved easely that $d(x,y)\leq 2 ...
0
votes
0answers
18 views

Problem with convergence of integral-type sums

We have two functions $f,g:[a,b] \rightarrow \mathbb R$ such that $f$ is integrable i the Riemann sense and $g$ is continuous. Let $(\pi_n)$, where: $\pi_n: ...
1
vote
2answers
65 views

question about a dot product

Let $$w(t)=\begin{cases} 0 &\text{ if } 0\le t\le \frac{1}{2}\\2t-1 &\text{ if }\frac{1}{2}\le t\le 1 \end{cases}$$ Is $$\langle f,g\rangle = \int_{0}^{1} w(t)f(t)g(t)dt$$ a dot ...
0
votes
1answer
21 views

Definition of a point $x$ in a Riemann sum.

$$\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta(x)$$ I am interested in what $x_k$ is. On stackexchange I have seen $x_k$ being defined as: $$x_k = \Delta(x)(k) + a$$ ...
2
votes
3answers
55 views

Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$

Let $a<b$ be real numbers. Let $f:[a,b]\to\mathbb{R}$ be a continuous non-negative function. Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$ Proof: Suppose for the sake of ...
3
votes
1answer
47 views

How prove this inequality $[f(x)]^2+[f'(x)]^2\le \max{(A,B)}$

let $f(x)$ be two derivable on $R$,give the two postive numbers $A,B$ and such $$[f(x)]^2\le A$$ $$[f'(x)]^2+[f''(x)]^2\le B$$ show that $$[f(x)]^2+[f'(x)]^2\le \max{(A,B)},\forall x\in ...
2
votes
1answer
40 views

Function holomorphic except for real line and continuous everywhere is entire

1) I've already shown that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic everywhere except for a single point and if it continuous on whole $\mathbb{C}$ then it is entire. It was quite easy. ...
0
votes
0answers
7 views

How can I verify that the kernel of Riesz transform is a fourier multiplier?

So I'm reading 'Wdighted Norm Inequalities and Related Topics' by Jose Garcia-Cuerva and Jose L.Rubio De Francia. In this book the definition of Fourier multiplier for $L^p(\mathbb{R}^n)$ is defined ...
0
votes
0answers
23 views

Prove the convergence of an (improper) integral

I have proved the following: Let $t\geq 0$ and $\alpha \in (-1/2, 1/2)$ given. Then there exists a $\gamma \in (1,2)$ such that $$ \int_0^t \int_{0}^{s'} \frac{((t-s)^{\alpha}- ...
0
votes
1answer
17 views

Residue Formula in complex analysis

I understand the residue formula but I just can't understand the cancelling down of $$ \operatorname{res}_{z=z_1} (f)= \lim \limits_{z \to z_1}(z-z_1) \frac {z^2}{z^4+1} = \frac {z_1^2}{4z_1^3}.$$ ...
2
votes
1answer
39 views

Finite rank volterra operator

I am wondering when a Volterra integral operator $V_K:L_2(0,1)\to L_2(0,1)$ is a finite rank operator: $$V_Kf=\int_0^xK(x,y)f(y)dy$$ thanks in advance for your help
0
votes
1answer
20 views

What is the significance of incompatible coordinate charts for a manifold?

For reference, here is my definition of a "manifold". A $\,C^\infty$ manifold is a topological manifold together with all the admissible charts of some $C^\infty$ atlas. When considering the ...
1
vote
1answer
31 views

Invariant subspace of self-adjoint operator

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
0
votes
0answers
11 views

center of gravity octans spehere

I have a center of gravity point $\overline{x_i}:= (\int_B x_i\rho(x)dx)/(\int_B\rho(x) dx)$ i=1,2,3. I have to count center of gravity of sphere octans, where B:={x $\in $[0,$\infty]^3 :|x| \le R $ ...
0
votes
1answer
27 views

How to prove $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = (n+1)^{p+1}-1$?

This is the solution $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = \sum_{k=1}^{p+1} \binom{p+1}{k} \sum_{l=1}^{n}l^{p+1-k} = \sum_{l=1}^{n}(l+1)^{p+1}-l^{p+1} = (n+1)^{p+1}-1$ ? With $S_{n}^{p} = ...
0
votes
1answer
14 views

On existence of extreme points of special type of non-empty closed convex sets of $\mathbb R^n$ [closed]

Let $A \subseteq \mathbb R^n $ be a non-empty closed(w.r.t. usual Euclidean metric of $\mathbb R^n$) convex set such that for some $x \in \mathbb R^n$ and $r>0$ , $B(x,r) \cap A=\phi$ , then must ...
2
votes
0answers
49 views

Differential one-forms and change of coordinates

Consider two differential one forms: $$\omega=\sum_{i=1}^N \omega_i dx^i$$ $$\omega'=\sum_{i=1}^N \omega'_i dx'^i$$ As I recall from my analysis courses, the symbols $dx$ are a particular notation ...
0
votes
1answer
23 views

Expansion into partial fractions

I've the following fractions given: $$\frac{a_0\cdot k_0}{b_0\cdot x + b_1\cdot x^2 + b_2 \cdot x^3 + b_4\cdot x^4+b_4\cdot x^4+b_5\cdot x^5}$$ $$\frac{a_0\cdot k_0}{b_0\cdot x^2 + b_1\cdot x^3 + ...
1
vote
1answer
38 views

Guidelines for choosing integrand to get a sum.

The idea was to find: $$\sum_{n=1}^{\infty} \frac{\coth(n\pi)}{n^3}$$ As you see in the solution, they conveniently choose a $f(z)$ they chose: $$f(z) = \frac{\pi \cot(\pi z)\coth(\pi z)}{z^3}$$ ...
0
votes
0answers
29 views

Why is taking the sum of residues required?

As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{N} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the ...
0
votes
2answers
65 views

Convention for dummy variables [duplicate]

I have just been introduced to the topic of integration. My teacher said: $$\int_{a}^{b} f(x) dx$$ Is the same thing as: $$\int_{a}^{b} f(\alpha) d\alpha$$ I asked her why, and she said "it is a ...